Modern Trends in Algebraic Graph Theory...(di)graph Gin terms of jVj, r, and the Laplacian...

Modern Trends inAlgebraic Graph Theory

An International Conference

Conference Programwith Abstracts

Villanova UniversityJune 2-5, 2014

Modern Trends in Algebraic Graph Theory Conference Program

Contents

Program Schedule 2

June 2 Monday 2

June 3 Tuesday 4

June 4 Wednesday 6

June 5 Thursday 8

Abstracts 10

List of Participants 40

1


Program Schedule

Monday, June 2 2014Morning schedule

7:00–8:15 Breakfast (Dougherty Hall)

8:30–8:55 Official Welcome (Connelly Center Cinema)

9:00–12:45 Plenary Session (Connelly Center Cinema)

9:00–10:00 Using graphs and Laplacian eigenvalues to evaluate block designsRosemary A. Bailey

10:00–10:25 Coffee break (Connelly Center)

10:30–11:30 Two-graphs, primitivity and rigidityPeter Cameron

11:45–12:45 Spectral characterizations of distance-regularity of graphsEdwin van Dam

13:00–14:15 Lunch (Dougherty Hall)

Monday, June 2 2014Afternoon schedule

14:30–16:45 Refreshments (Room 115, Mendel Science Center)

14:30–17:20 Parallel Session I (Room 154, Mendel Science Center)

14:30–14:50 The history of the Spectral Excess theoremMiquel Angel Fiol

15:00–15:20 On graphs with three distinct eigenvaluesJack H. Koolen

15:30–15:50 An interlacing approach for bounding the sum of Laplacianeigenvalues of graphsAida Abiad

16:00–16:20 Spectral properties of simplicial rook graphsJason Vermette

16:30–16:50 A survey of cyclotomic eigenvalue problems in algebraiccombinatoricsAllen Herman

2


17:00–17:20 Spectra of graph transformationsAiping Deng

14:30–17:20 Parallel Session II (Room 213, Mendel Science Center)

14:30–14:50 Polyadic constacyclic codesYun Fan

15:00–15:20 Prefix-reversal Gray codesElena Konstantinova

15:30–15:50 Friendship Theorem for hypergraphs: a constructionLeif Jørgensen

16:00–16:20 Large sets of q-analogs of designsReinhard Laue

16:30–16:50 Flag-transitive symmetric designs with (r, λ) = 1 and alternatingsocleShenglin Zhou

17:00–17:20 p-Ranks of quasi-symmetric designs and standard modules ofcoherent configurationsAkihide Hanaki

18:00–19:15 Dinner (Dougherty Hall)

3


Tuesday, June 3 2014Morning schedule



9:00–10:00 Weakly locally projective actionsAlexander A. Ivanov


10:30–11:30 Algebraic map theoryGareth A. Jones

11:45–12:45 Hypergroups, association schemes, buildingsPaul-Herman Zieschang


Tuesday, June 3 2014Afternoon schedule



14:30–14:50 Two-fold isomorphismsJosef Lauri

15:00–15:20 Half-arc transitive group actions with a small number of alternetsKlavdija Kutnar

15:30–15:50 Semiregular subgroups of transitive permutation groupsDragan Marušič

16:00–16:20 Cyclotomic association schemes and their fusion and fission schemesSung-Yell Song

16:30–16:50 Automorphisms of association schemes whose relations are ofvalency one or threeMitsugu Hirasaka

17:00–17:20 Sandpile groups of generalized de Bruijn and Kautz graphs andcirculant matrices over finite fieldsDmitrii V. Pasechnik

4



14:30–14:50 Distance-regular extensions of strongly regular graphs witheigenvalue 3Alexandr Makhnev

15:00–15:20 Combinatorial schemes in algebraic algorithmsNitin Saxena

15:30–15:50 On the connectedness of the complement of a ball in distance-regular graphsSebastion Cioaba

16:00–16:20 Association schemes on general measure spaces and zero-dimensional Abelian groupsAlexander Barg

16:30–16:50 Graph homotopy, ideals of finite varieties and a surprising dualityWilliam J. Martin

17:00–17:20 Equiangular lines in Euclidean spacesGary Greaves


5


Wednesday, June 4 2014Morning schedule



9:00–10:00 Q-polynomial association schemesJason Williford


10:30–11:30 Symmetry versus regularityLászló Babai



12:00–12:20 On non-abelian Schur groupsAndrey Vasil’ev

12:30–12:50 On characterization of the Grassmann graphs J2(2d, d), d ≥ 3,by their intersection arraysAlexander Gavrilyuk


12:00–12:20 Circulant graphs, nonlinear loop transversal codes andnonassociative loopsJonathan D. H. Smith

12:30–12:50 Latin squares, determinants and permanentsKenneth Johnson


Wednesday, June 4 2014Afternoon schedule



14:30–14:50 Automorphism groups of Schur rings over cyclic groups ofprime-power orderReinhard Pöschel

6


15:00–15:20 On circulant graphs that are isomorphic to Cayley graphs ofmore than one abelian groupTed Dobson

15:30–15:50 On groups all of whose undirected Cayley graphs of boundedvalency are integralIstván Kovács

16:00–16:20 Some recent advances in analytic enumeration of circulant graphsValery Liskovets

16:30–16:50 Coset closure of a circulant S-ring and the schurity problemIlia Ponomarenko

17:00–17:20 Geometric clique structures in association schemesJohn Wilmes


14:30–14:50 The Erdős-Ko-Rado Theorem: an algebraic perspectiveKaren Meagher

15:00–15:20 Operations on graphs increasing some graph parametersAlexander Kelmans

15:30–15:50 On algebraic constructions of graphs without small cycles andcommutative diagrams and their applicationsVasya Ustimenko

16:00–16:20 Properties of some algebraically defined digraphsAleksandr Kodess

16:30–16:50 On some cycles in Wenger graphsYe Wang

17:00–17:20 Generalized quadrangles and algebraically defined graphsBrian Kronenthal

19:00–22:00 Banquet (Villanova Room, Connelly Center)Bar opens at 19:00; dinner is served at 19:30.Prepaid tickets are required for admission.Attire is casual. Seating is not prearranged.

7


Thursday, June 5 2014Morning schedule



9:00–10:00 Using computational group theoryAlexander Hulpke



10:30–10:50 On highly regular strongly regular graphsChristian Pech

11:00–11:20 Numbers of induced subgraphs in strongly regular graphsKristína Kováčiková

11:30–11:50 The extendability of matchings in strongly regular graphsWeiqiang Li

12:00–12:20 Graphs similar to strongly regular graphsKatarína Tureková

12:30–12:50 On strongly regular graphs attaining the claw boundMartin Mačaj


10:30–10:50 Bidirected edge-maximality of power graphs of finitecyclic groupsBrian Curtin

11:00–11:20 Embedding factorizations of hypergraphsMike Newman

11:30–11:50 Intrinsic metrics and geometry on graphsDouglas Klein

12:00–12:20 Product sets and algebraic graph theoryMatt Devos


8


Thursday, June 5 2014Afternoon schedule



14:30–14:50 Group actions on posets and cartesian productsAnton Betten

15:00–15:20 On cliques in edge-regular graphsLeonard Soicher

15:30–15:50 Three-class association schemes related to a constructionby MathonSven Reichard

16:00–16:20 New rich infinite families of directed strongly regular graphsŠtefan Gyürki

16:30–16:50 New families of non-Schurian association schemes related toHeisenberg groups and biaffine planesMikhail Klin

17:00–17:20 Constructive enumeration of coherent configurations of small orderMatan Ziv-Av


14:30–14:50 Billiard Arrays and finite-dimensional irreducible Uq(sl2)-modulesPaul Terwilliger

15:00–15:20 Finite-dimensional irreducible modules for an even subalgebraof Uq(sl2)Alison Gordon Lynch

15:30–15:50 Tridiagonal pairs of q-Racah type and the quantum envelopingalgebra Uq(sl2)Sarah Bockting Conrad

16:00–16:20 Coxeter table algebrasAndrew Wang

16:30–16:50 Orthogonal polarity graphs and Sidon SetsMichael Tait

17:00–17:20 Multiplicities of irreducible characters of table algebras andapplications to association schemesBangteng Xu


9


AbstractsSpeaker: Aida Abiad ([email protected]) Tilburg UniversityCoauthors: Miquel Angel Fiol, Willem Haemers, and Guillem PerarnauTitle: An interlacing approach for bounding the sum of Laplacian eigenval-ues of graphsAbstract: We apply eigenvalue interlacing techniques for obtaining lower andupper bounds for the sums of Laplacian eigenvalues of graphs, and charac-terize equality. This leads to generalizations of, and variations on, theoremsby Grone, and Grone & Merris. As a consequence we obtain inequalitiesinvolving bounds for some well-known parameters of a graph, such as edge-connectivity, and the isoperimetric number.

Speaker: László Babai ([email protected]) University of ChicagoTitle: Symmetry versus regularityAbstract: Our theme is the seeming paradox that regularity constraints oftenseverely limit the number and the structure of symmetries, possibly withclassifiable exceptions. A recent example: If X is a Steiner 2-designs withn points then |Aut(X)| ≤ nO(logn) (B-Wilmes and Chen-Sun-Teng, 2013).This result represents progress toward the speaker’s long-held conjecture:Conjecture 1. For every ε > 0, if X is a strongly regular graph (SRG) withn ≥ n(ε) vertices then |Aut(X)| ≤ exp(nε), with known (easy) exceptions.The conjecture is known to hold for all ε > 9/37 (Xi Chen, Xiaorui Sun, and

Shang-Hua Teng, 2013). Furthermore, the automorphism groups of SRGssatisfy a strong structural constraint: their thickness is O(log2 n/ log log n),with the same known exceptions (B, 2014). (The thickness θ(G) of a groupG is the largest t for which the alternating group At is a quotient of asubgroup of G.) This settles, in a surprisingly strong form, a relaxation ofConjecture 1, potentially relevant to the Graph Isomorphism problem. Theresult has recently (May 2014) been used by Pyber to give a long-soughtelementary proof of a quasipolynomial (exp((log n)C)) bound on the order ofrank-3 permutation groups (with the corresponding known exceptions). Thethickness bound uses spectral arguments and an old combinatorial lemmaon permutations by the late Ákos Seress (1958 – 2013) and the speaker.Coherent configurations (CCs) are “directed association schemes.” Xiaorui

Sun and John Wilmes have recently (2014) made progress on the followingextension of Conjecture 1 (see Wilmes’s talk).Conjecture 2. If X is a primitive CC with n ≥ n(ε) vertices and |Aut(X)| >exp(nε) then Aut(X) is a primitive permutation group and X its orbital

10


configuration (and therefore X is known by a result of Cameron (1980)).

Speaker: R. A. Bailey ([email protected]) University of St AndrewsTitle: Using graphs and Laplacian eigenvalues to evaluate block designsAbstract: Consider an experiment to compare v treatments in b blocks ofsize k. Statisticians use various criteria to decide which design is best. Itturns out that most of these criteria are defined by the Laplacian eigenvaluesof one of the two graphs defined by the block design: the Levi graph, whichhas v+ b vertices, or the concurrence graph, which has v vertices. The alge-braic approach shows that sometimes all the criteria prefer highly symmetricdesigns but sometimes they favour very different ones.

Speaker: Alexander Barg ([email protected]) University ofMarylandCoauthor: Maxim SkriganovTitle: Association schemes on general measure spaces and zero-dimensionalAbelian groupsAbstract: The main purpose of this talk is to define and study associationschemes on infinite sets. Direct extensions of the definition to infinite setsencounter some problems even in the case of countable sets, for instance,countable discrete Abelian groups. In an attempt to resolve these difficulties,we define association schemes on arbitrary, possibly uncountable sets with ameasure. We construct large classes of examples of schemes on topologicalzero-dimensional Abelian groups, for instance, Cantor-type groups or thegroups of p-adic numbers. Employing duality theory and the machinery ofharmonic analysis, we compute their eigenvalues and intersection numbers.For one class of schemes constructed in this work, the eigenvalues coincide

with values of orthogonal function systems on zero-dimensional groups. Weobserve that these functions have the properties of wavelet bases on thegroup, establishing a seemingly new link between algebraic combinatoricsand harmonic analysis. Preprint: arXiv:1310.5359.

Speaker: Anton Betten ([email protected]) Colorado StateUniversityTitle: Group actions on posets and cartesian productsAbstract: Classifying combinatorial objects like graphs and other thingsrequires us to compute orbits of groups acting on partially ordered sets.There is a long history of algorithms for this purpose. The term ‘orderly

11


generation’ was coined for exactly this purpose and the roots of this goback to people like Read and Faradjev. More recent work has been doneby McKay. We take a look at the theory behind these algorithms, in orderto better understand their inner workings. In doing so, we will encounter alittle lemma on permutation groups acting on subsets of cartesian products,potentially intransitive. This lemma seems to be part of the mathematicalfolklore.

Speaker: Sarah Bockting-Conrad ([email protected]) Uni-versity of WisconsinTitle: Tridiagonal pairs of q-Racah type and the quantum enveloping algebraUq(sl2)

Abstract: In this talk we explore a connection between tridiagonal pairs of q-Racah type and the quantum enveloping algebra Uq(sl2). Given a tridiagonalpair A, A∗ of q-Racah type, we introduce linear transformations ψ : V → V ,K : V → V , and B : V → V which act on the split decompositions of V inan attractive way. Using ψ, K, B we obtain two Uq(sl2)-module structureson V . For each of the Uq(sl2)-module structures, we compute the action ofthe Casimir element on V . We show that these two actions agree. Usingthis fact, we express ψ as a rational function of K±1, B±1 in several ways.Eliminating ψ from these equations we find that K and B are related by aquadratic equation.

Speaker: Peter Cameron ([email protected]) University of St An-drewsTitle: Two-graphs, primitivity and rigidityAbstract: Two-graphs were introduced by Graham Higman to construct Con-way’s third group, and studied by Jaap Seidel in connection with equiangularlines in Euclidean space. More recently, Misha Gromov proposed a methodfor finding bounds in a problem about convex sets in Euclidean space, whichinvolves certain evaluations of which the first one concerns two-graphs; DanKrál´ and others have used this to find new bounds for these constants.Since the Classification of Finite Simple Groups, we know that primitive

permutation groups (other than symmetric and alternating groups) are rel-atively small. I will give a new example of such a result, showing that if anon-trivial two-graph has a primitive automorphism group, then some graphin the corresponding switching class has trivial automorphism group, apartfrom a finite number of exceptions.

12


Speaker: Sebastion Cioaba ([email protected]) University ofDelawareCoauthor: Jack KoolenTitle: On the connectedness of the complement of a ball in distance-regulargraphsAbstract: Punch a not too big hole in a distance-regular graph. The resultwill still be connected.In different words, an important property of strongly regular graphs is that

the second subconstituent of any primitive strongly regular graph is alwaysconnected. Brouwer asked to what extent this statement can be generalizedto distance-regular graphs. In this paper, we show that if γ is any vertex ofa distance-regular graph Γ and t is the index where the standard sequencecorresponding to the second largest eigenvalue of Γ changes sign, then thesubgraph induced by the vertices at distance at least t from γ, is connected.

Speaker: Brian Curtin ([email protected]) University of SouthFloridaCoauthor: G. R. PourgholiTitle: Bidirected edge-maximality of power graphs of finite cyclic groupsAbstract: The directed power graph of a finite group is the graph whosevertices are the group elements and an edge from one element to another ifthe second is a power of the first. We discuss our recent work showing thatamong all finite groups of any given order, the cyclic group of that orderhas the maximum number of pairs of oppositely directed edges in its powergraph.

Speaker: Edwin van Dam ([email protected]), Tilburg UniversityTitle: Spectral characterizations of distance-regularity of graphsAbstract: The eigenvalues of the adjacency matrix of a graph contain a lot— but not always all — information on the structure of the graph. In thistalk, we will dive deeper into the case of distance-regular graphs. We willgive an overview of when distance-regularity is determined by the eigenval-ues (and when it is not). We will for example see how systems of orthogonalpolynomials can help to recognize distance-regular graphs from their eigen-values and a little extra information through the ‘spectral excess theorem’.We then discuss how these methods and ideas led to the construction of thetwisted Grassmann graphs, a family of distance-regular graphs that havethe same spectrum as certain Grassmann graphs. These twisted graphs arecurrently the only known family of distance-regular graphs with unbounded

13


diameter that are not vertex-transitive. We also discuss some recent resultsthat involve the Laplacian matrix, the so-called preintersection numbers, anddistance-regular Taylor graphs.

Speaker: Aiping Deng ([email protected]) Donghua UniversityCoauthor: Alexander KelmansTitle: Spectra of graph transformationsAbstract: Given a simple graph G with vertex set V (G) = V and edge setE(G) = E, let Gl be the line graph and Gc the complement of G. Let G0

be the graph with V (G0) = V and no edges, G1 the complete graph withvertex set V , G+ = G and G− = Gc. Let B(G) (Bc(G)) be the graph withvertex set V ∪ E and such that (v, e) is an edge in B(G) (resp., in Bc(G))if and only if v ∈ V , e ∈ E and vertex v is incident (resp., not incident)to edge e in G. Given x, y, z ∈ 0, 1,+,−, the xyz-transformation Gxyz ofG is the graph with vertex set V (Gxyz) = V ∪ E and edge set E(Gxyz) =E(Gx)∪E((Gl)y)∪E(W ), whereW = B(G) if z = +, W = Bc(G) if z = −,W is the graph with V (W ) = V ∪ E and no edges if z = 0, and W is thecomplete bipartitie graph with parts V and E if z = 1. The definition ofxyz-transformation is valid for general graphs when x, y∩ −, 1 = ∅. Wealso extend this notion to simple digraphs and digraphs with possible loops.We obtain the Laplacian (adjacency) characteristic polynomials and some

other Laplacian parameters of every xyz-transformation of a simple r-regular(di)graph G in terms of |V |, r, and the Laplacian (adjacency) spectrum ofG. Similar results are obtained for digraphs with possible loops as well asfor some non-regular digraphs. We also give constructions of non-isomorphicand Laplacian (adjacency) cospectral digraphs using xyz-transformations.

Speaker: Matt Devos ([email protected]) Simon Fraser UniversityTitle: Product sets and algebraic graph theoryAbstract: A classical problem in additive number theory is to determine thestructure of those pairs of sets A,B in a (multiplicative) group for which theproduct set AB is small. The most extreme notion of small which remainsinteresting is the case when |AB| < |A| + |B|. Here the complete structurewas determined by Vosper for groups of prime order and by Kemperman forabelian groups. In a very lengthy project completed just over a year ago,we solved this for arbitrary groups. This talk will reveal why this problemis secretly one in extremal algebraic graph theory, and give an indication ofhow it is proved. We then turn our attention to some related questions fromthis realm.

14


Speaker: Ted Dobson ([email protected]) Mississippi State Uni-versity and University of PrimorskaCoauthor: Jay MorrisTitle: On circulant graphs that are isomorphic to Cayley graphs of morethan one abelian groupAbstract: We show that for certain integers n, the problem of whether ornot a circulant digraph Γ of order n is also isomorphic to a Cayley digraphof some other abelian group G of order n reduces to the question of whetheror not a natural subgroup of the full automorphism group contains morethan one regular abelian group up to isomorphism (as opposed to the fullautomorphism group). A necessary and sufficient condition is then givenfor such circulants to be isomorphic to Cayley digraphs of more than oneabelian group. A generalization of a permutation group theoretic result ofMuzychuk is one of the main tools.

Speaker: Yun Fan ([email protected]) Central China Normal Uni-versityTitle: Polyadic constacyclic codesAbstract: Necessary and sufficient conditions of the existence of polyadicconstacyclic codes are reported in terms of p-valuations; the main ideas tosolve the question are described. Some consequences are derived, and someexamples are exhibited.

Speaker: M. A. Fiol ([email protected]) Universitat Politècnica deCatalunyaTitle: The history of the Spectral Excess theoremAbstract: Let G be a (connected) regular graph with d+ 1 distinct eigenval-ues. The excess of a vertex u is the number of vertices at maximum distanced from u, whereas the spectral excess is a number which can be computedfrom the spectrum of G (eigenvalues and multiplicities). The spectral excesstheorem, due to Garriga and the speaker (1997), states that G is distance-regular if and only if the excess of every vertex is what it should be. Moreprecisely, it states that, for any regular graph, the average excess is at mostthe spectral excess, and equality occurs if and only if G is distance-regular.The aim of this talk is to give an overview of the history of this result: whichprevious results inspired it; its different versions; and some of its knownapplications.

15


Speaker: Alexander Gavrilyuk ([email protected])Tohoku UniversityCoauthor: Jack H. KoolenTitle: On characterization of the Grassmann graphs J2(2d, d), d ≥ 3, bytheir intersection arraysAbstract: The Grassmann graph Jq(v, d) is a graph defined on the set ofd-dimensional subspaces of a v-dimensional vector space over the finite fieldFq, with two subspaces U and W being adjacent if dim(U ∩W ) = d− 1.In 1995, K. Metsch showed that a distance-regular graph with intersection

array of Jq(v, d) is indeed Jq(v, d), if v 6= 2d, v 6= 2d ± 1, (v 6= 2d ± 2 ifq ∈ 2, 3), and (v 6= 2d± 3 if q = 2).In this work, we show that the Grassmann graph J2(2d, d) with diameter

d ≥ 3 is characterized by its intersection array.

Speaker: Gary Greaves ([email protected]) Tohoku UniversityCoauthors: J. Koolen, A. Munemasa, and F. SzöllősiTitle: Equiangular lines in Euclidean spacesAbstract: Given some dimension d, what is the maximum number of lines inRd such that the angle between any pair of lines is constant? This classicalproblem has recently enjoyed a renewed interest due to the current atten-tion the quantum physics community is giving to its complex analogue. Wewill report on some new developments of the theory of equiangular lines inEuclidean spaces. Among other things, we will present improvements to twolong standing upper bounds for equiangular lines in dimensions 14 and 16.As a consequence, we will also establish the nonexistence of certain regulargraphs with four distinct eigenvalues.

Speaker: Štefan Gyürki ([email protected]) Slovak University of Tech-nologyCoauthor: Mikhail KlinTitle: New rich infinite families of directed strongly regular graphsAbstract: The notion of a directed strongly regular graph was introducedby Duval as a possible generalization of classical strongly regular graphs. Adirected strongly regular graph (DSRG) with parameters (n, k, t, λ, µ) is aregular directed graph Γ on n vertices with valency k, such that every vertexis incident with t undirected edges, and the number of paths of length 2directed from a vertex x to a vertex y is λ if there is an arc from x to y, andµ otherwise. We present four infinite families of DSRGs with parameter sets(2n2, 3n−2, 2n−1, n−1, 3), (2n2, 4n−2, 2n+2, n+2, 6), (3n2, 4n−2, 2n, n, 4),

16


(3n2, 6n− 2, 2n+ 6, n+ 6, 10). Our constructions correspond to an arbitraryLatin square L of order n. Vertices of a related graph Γ(L) are formed bydirect product of sets of rows, columns and sets of cardinality 2 or 3, whileadjacency in Γ(L) is described (for each of 4 kinds) in terms of L.Extensive computer algebra experimentation allows us to guess about the

conditions for the isomorphism of corresponding DSRGs Γ(L) versus a suit-able type of equivalence of Latin squares L. Relying on the analysis ofthese computer results we will briefly discuss our attempts to formulateplausible conjectures (for each of the four families) about the number ofnon-isomorphic DSRGs Γ(L).

Speaker: Akihide Hanaki ([email protected]) Shinshu UniversityTitle: p-Ranks of quasi-symmetric designs and standard modules of coherentconfigurationsAbstract: It is known that incidence matrices of some designs with sameparameter have different p-ranks, i.e. ranks over a field of characteristic p.For example, there are 80 non-isomorphic 2-(15, 3, 1)-designs and their 2-ranks are 11, 12, 13, 14, and 15. A 2-(v, l, 1)-design is said to be quasi-symmetric. From a quasi-symmetric design, naturally, we can constructa coherent configuration. We will see that the structure of the modularstandard module determine the p-rank. Also we will determine p-modularstandard module of coherent configurations defined by 2-(15, 3, 1)-designsfor p = 2, 3, 5 and 7.

Speaker: Allen Herman ([email protected]) University of ReginaTitle: A survey of cyclotomic eigenvalue problems in algebraic combinatoricsAbstract: A complex number λ is cyclotomic if it is contained in the fieldQ(ζn), where ζn denotes an n-th root of unity for some positive integern. A question in the book of Bannai and Ito asks if all the eigenvalues offinite commutative association schemes are cyclotomic. Since then severalclosely-related problems have emerged. We will report the latest ideas onthese problems from both the positive and negative directions. From thepositive direction, the best known answer for some time has been yes forSchurian configurations, and for P-polynomial schemes with rank at least34. From the negative direction, a character-table characterization of thefield of Krein parameters of a table algebra yields a necessary condition forit to have non-cyclotomic eigenvalues. We will conclude the talk by givingthe smallest examples of connected regular graphs (on 9 and 10 vertices)with non-cyclotomic eigenvalues.

17


Speaker: Mitsugu Hirasaka ([email protected]) Pusan NationalUniversityCoauthor: Jeong Rye ParkTitle: Automorphisms of association schemes whose relations are of valencyone or threeAbstract: In this talk we deal with association schemes whose relations areof valency one or three, and aim to find a non-trivial automorphism of suchan association scheme under some assumptions on intersection numbers.

Speaker: Alexander Hulpke ([email protected]) ColoradoState UniversityTitle: Using computational group theoryAbstract: Groups and similar structures are often used to construct com-binatorial structures such as graphs. Concrete constructions, or analysis ofthe combinatorial structures then often require structural information onthe algebraic side or construction of algebraic subobjects. Motivated by thisduality I will survey what computational group theory, in particular the sys-tem GAP, can do and how to formulate problems in a way to increase thelikeliness of getting a (usable) result.

Speaker: Alexander A. Ivanov ([email protected]) ImperialCollegeTitle: Weakly locally projective actionsAbstract: Let ∆ be an undirected, connected locally finite graph with auto-morphism group G. For a vertex x of ∆, denote by G(x)∆(x) the permutationgroup induced by the action of the stabilizer G(x) on the neighbor set ∆(x)of x. We call the action of G on ∆ is weakly locally projective if for everyvertex x ∈ ∆, there is a positive integer nx and a prime power qx such that

|∆(x)| = (qnxx − 1)/(qx − 1) and Lnx(qx) 6 G(x)∆(x) 6 PΓL(nx, qx).

Here Lnx(qx) is considered as a doubly transitive permutation group of degree|∆(x)| and PΓL(nx, qx) is the normalizer of Lnx(qx) in the symmetric groupon ∆(x). If a weakly locally projective action is also vertex-transitive it issaid to be locally projective. Since every weakly locally projective action isedge-transitive it is easy to see that either:(a) the action is locally projective, and there is a pair (n, q) such that

nx = n and qx = q for every x ∈ ∆, or(b) there is bipartition ∆ = ∆1 ∪∆2 of ∆ and a quadruple of parameters

(n1, q1;n2, q2) such that nx = ni, qx = qi whenever x ∈ ∆i.

18


Examples of (weakly) locally projective actions come from classical groupsincluding Ln(q) = An−1(q), Sp2n(q) = Bn(q), and G2(q), as well as fromsporadic simple groups including M22, M23, M24, He, Co2, Co1, J4, BMand M . Thus, among other things the study of weakly locally projectiveactions actions might lead to a theory with no gap between classical andsporadic groups. A possible layout of such a theory will be discussed.

Speaker: Kenneth Johnson ([email protected]) Pennsylvania State Uni-versity at AbingtonCoauthors: I. Wanless and D. DonovanTitle: Latin squares, determinants and permanentsAbstract: The determinant of the matrix arising from the operation of rightdivision in a group lies at the foundation of group representation theory.More precisely, if G = g1, ..., gn is a group, and xg1 , ..., xgn is a set ofcommuting variables the group matrix XG is the n×n matrix whose (i, j)th

entry is xgig−1j . The group determinant Θ(G) is the determinant of XG. The

factorisation of Θ(G) led to the discovery of group characters. It can beshown that Θ(G) determines G uniquely. It can also be shown that the per-manent of XG determines G uniquely.To any n×n latin square there is associated a determinant (or permanent)

by replacing the elements in the square by variables. Squares L1 and L2

have the same determinant if either they are isomorphic or L1 is the trans-pose of L2. Call the corresponding equivalence relation trisotopy. It is nottrue that trisotopy classes of squares are determined by their determinant orpermanent, but cases where this occurs seem to be sparse.I will talk about the information in the determinant and/or the permanent

of a latin square, and relate this to the algebraic and combinatorial proper-ties of the square. There is a connection with graph theory in that the “latinsquare matrix” can be associated with random walks on the analog of theCayley graphs of the corresponding quasigroups.

Speaker: Gareth A. Jones ([email protected]) University ofSouthamptonTitle: Algebraic map theoryAbstract: My talk will be a survey of algebraic map theory, that is, a group-theoretic approach to embeddings of graphs in surfaces (assumed to be com-pact and oriented, for simplicity). I shall concentrate on two related researchtopics. One is the study of regular maps: these are the most symmetric

19


embeddings of various classes of arc-transitive graphs (complete, completebipartite, etc), where various techniques from finite group theory yield classi-fications. The other is the use of bipartite graph embeddings (called dessinsd’enfants) to provide a bridge between the theories of Riemann surfacesand of algebraic number fields, with regular dessins recently proved to give afaithful representation of an important and mysterious profinite group calledthe absolute Galois group.

Speaker: Leif Jørgensen ([email protected]) Aalborg UniversityCoauthor: Anita SillasenTitle: Friendship Theorem for hypergraphs: a constructionAbstract: V. Sós (1976) considered 3-uniform hypergraphs with the followingproperty: for any three vertices x, y, z there is a unique vertex w so thatw, x, y, w, x, z and w, y, z are hyperedges. Such graphs are calledfriendship hypergraphs. Sós observed that a friendship hypergraph with avertex u so that u, x, y is a hyperedge for all other x and y is equivalentto a Steiner triple system. Five friendship hypergraphs with no such vertexwere found by Hartke and Vandenbussche (2008). We construct an infinitefamily of friendship hypergraphs and a few examples not in that family. Allof our friendship hypergraphs are partially balanced designs based on anassociation scheme (a hypercube for the infinite family).

Speaker: Alexander Kelmans ([email protected]) Uni-versity of Puerto Rico, Rutgers UniversityTitle: Operations on graphs increasing some graph parametersAbstract: In this talk we will discuss some of our old and recent results onpartial orders on the set Gmn of graphs with n vertices and m edges and someoperations on graphs within Gmn that are monotone with respect to thesepartial orders. The partial orders under consideration include those relatedwith the Laplacian and adjacency characteristic polynomials, the coefficientsof the Laplacian polynomial, the chromatic polynomial, the matching poly-nomial, the independence polynomial of a graph as well as with the numbersof spanning subgraphs of certain types and some probabilistic characteristicsof graphs with randomly deleted edges. The results on the monotonicity ofcertain operations with respect to some graph parameters or posets turnedout to be useful in the study of the corresponding combinatorial optimiza-tion problems and in attempts to find the structure of graphs having someextremal properties such as graphs in Gmn with the maximum (or minimum)number of connected spanning subgraphs of a given size or graph reliability.

20


Speaker: Doug Klein ([email protected]) Texas A&M University atGalvestonTitle: Intrinsic metrics and geometry on graphsAbstract: Possible intrinsic metrics on a finite connected graph G are con-sidered. For longer-wavelength waves on G, squares of wave-amplitude dif-ferences on two vertices might lead to an intersite “distance”. The numberof spanning 2-component acyclic subgraphs with vertices i & j in separatecomponents might give a distance between i & j (as the farther i & j areapart, the more intervening edges there are to delete in a spanning tree). Forrandom walks on G, the probability of a walk returning to a starting vertex ibefore reaching vertex j could vary inversely with a distance between i & j.And the effective electrical resistance between i & j might be a metric. Someresults for such candidate metrics are noted. Uses to measure cyclicity, orcentrality, or foldedness seem possible, as do invariants analogous to classicalEuclidean-geometric quantities.

Speaker: Mikhail Klin ([email protected]) Ben-Gurion University of theNegevCoauthor: Štefan GyürkiTitle: New families of non-Schurian association schemes related to Heisen-berg groups and biaffine planesAbstract: For p ≥ 3 a prime, let H(p) be the Heisenberg group of orderp3, and let B(p) be the classical biaffine plane of order p (i.e., Desarguesianaffine plane of order p with one parallel class of lines removed). The groupH(p) acts naturally (with two orbits of length p2) on the set Ω(p) of pointsand lines of B(p).Relying on extensive computer algebra experimentation, the authors an-

nounced in 2012 the existence of four new families Mi(p), i ∈ [1, 4], of non-Schurian association schemes on 2p2 vertices, with respective automorphismgroups of order p3, 2p3, 2p3, 8p3. The rank of these schemes grows linearlywith p. Many interesting properties of these schemes have been justifiedtheoretically, while some properties still appear as plausible conjectures.Recently, we discovered the existence of two other new families F (p) and

S(p) of non-Schurian schemes of rank 5 and 6 respectively, which are also in-variant with respect to (H(p),Ω(p)), though the order of the automorphismgroups of these schemes grows with p more quickly. Some information aboutthese new schemes (for example, their intersection matrices and spectra) willbe considered, together with a discussion about ongoing attempts to reachfull justification of the properties of Mi(p), considering them in conjunction

21


with the new schemes of rank 5 and 6.This research was supported at Matej Bel University (Slovakia) by the

European Social Fund, ITMS code: 26110230082.

Speaker: Aleksandr Kodess ([email protected]) University ofDelawareCoauthor: Felix LazebnikTitle: Properties of some algebraically defined digraphsFor a prime power q and a vector function f = (f1, . . . , fl):F2

q → Flq, wedefine a directed graph D = D(q; f) as follows: the vertex set V of D isFl+1q , and there is an arc from (x1, . . . , xl+1) ∈ V to (y1, . . . , yl+1) ∈ V ifxi + yi = fi−1(x1, y1) for all i, 2 ≤ i ≤ l+ 1. We study the strong connectiv-ity of such digraphs and describe the structure and number of their strongcomponents depending on certain properties of the image of f . For the casewhen f :F2

q → Fq and f(x, y) = xmyn for some integers 1 ≤ m,n ≤ q − 1, wedenote D(q; f) by D(q;m,n). We also study the question of isomorphism ofdigraphs D(q;m1, n1) and D(q;m2, n2), and we conjecture that the digraphsare isomorphic if and only if m2 = km1 and n2 = kn1 for some k coprimewith q − 1.

Speaker: Elena Konstantinova ([email protected]) Sobolev In-stitute of Mathematics, Russian Academy of Sciences, NovosibirskCoauthor: Alexey MedvedevTitle: Prefix-reversal Gray codesAbstract: Two scenarios are known to get prefix-reversal Gray codes. Thefirst one was given by S. Zaks in 1984 [BIT, 24, 196-204], and the secondone was suggested by A. Williams and J. Sawada in 2013 [Electronic Notesin Discrete Math., 44, 357-362]. In this talk we consider other approachesfor getting prefix-reversal Gray codes.

Speaker: Jack H. Koolen ([email protected]) University of Scienceand Technology of China.Co-author: Ximing ChengTitle: On graphs with three distinct eigenvaluesAbstract: Connected graph with three distinct eigenvalues, which are thestrongly regular graphs have received a lot of attention. For the irregularcase they did not receive much attention. I will present some new results onthem.

22


Speaker: Kristína Kováčiková ([email protected])Comenius UniversityCoauthor: Martin MačajTitle: Numbers of induced subgraphs in strongly regular graphsAbstract: Let us fix a graph Γ. By PG we denote the number of occurrencesof graph G as an induced subgraph in Γ. Clearly, the values PK1 , PK2 andPK2

represent the numbers of vertices, edges and non-edges in Γ, respec-tively. If Γ is a strongly regular graph (SRG) with parameters (n, k, λ, µ),then it is known that the value PG of any graph G on at most three vertices isdetermined uniquely by these parameters. Unfortunately, with G spanningmore than 3 vertices, this nice property is no longer satisfied. An example ofsuch behavior are two non-isomorphic SRGs with parameter set (16, 6, 2, 2)and different values of PK4 .We study how the values of PG for all the graphs on t vertices interact. For

a triangle-free SRG we show that PG is determined by (n, k, λ, µ) for any Gon at most five vertices. When G is a graph on six vertices, PG depends alsoon the value PK3,3 .For the putative Moore graph with parameters (3250, 57, 0, 1), PG is de-

termined uniquely for any graph G on up to 9 vertices. For all graphs on 10vertices, the values PG are dependent only on the number of occurrences ofthe Petersen graph in this SRG.

Speaker: István Kovács ([email protected]) University of PrimorskaCoauthor: István EstélyiTitle: On groups all of whose undirected Cayley graphs of bounded valencyare integralAbstract: A finite group G is called Cayley integral if each undirected Caleygraph over G is integral, i.e., it has integral eigenvalues. The Cayley integralgroups have been determined by Kloster and Sander in the abelian case, andby Abdollahi and Jazaeri, and independently by Ahmady, Bell and Moharin the non-abelian case. In this talk we consider the class Gk of finite groupsall of whose undirected Cayley graphs of valency at most k are integral. Weprove that Gk consists of the Cayley integral groups if k ≥ 6; and the classesG4 and G5 are equal, and consist of: (1) the Cayley integral groups, (2) thegeneralized dicyclic groups Dic(Zn3 × Z6), where n ≥ 1.

23


Speaker: Brian Kronenthal ([email protected]) KutztownUniversityCoauthors: Felix Lazebnik and Jason WillifordTitle: Generalized quadrangles and algebraically defined graphsAbstract: In this talk, we will study generalized quadrangles from the per-spective of their point-line incidence graphs. In particular, the incidencegraphs of classical generalized quadrangles of odd prime power order q con-tain induced bipartite subgraphs that may be defined algebraically; indeed,defining partite sets P = Fq3 = L, we say vertices (a1, a2, a3) ∈ P and[x1, x2, x3] ∈ L are adjacent if and only if a2+x2 = a1x1 and a3+x3 = a1x1

2.This subgraph has girth eight. Of particular interest is whether it is possibleto alter these equations to create a nonisomorphic girth eight graph. Successcould illuminate a strategy for constructing new generalized quadrangles.

Speaker: Klavdija Kutnar ([email protected]) University of Pri-morskaCoauthors: Ademir Hujdurović and Dragan Marušič.Title: Half-arc transitive group actions with a small number of alternetsA graph X is said to be G-half-arc-transitive if G ≤ Aut(X) acts transitivelyon the set of vertices of X and on the set of edges of X but does not acttransitively on the set of arcs of X. Such graphs can be studied via corre-sponding alternets, that is, equivalence classes of the so-called reachabilityrelation, first introduced by Cameron, Praeger and Wormald in [Combina-torica 13 (1993), 377–396]. If the vertex sets of two adjacent alternets eithercoincide or have half of their vertices in common the graph is said to betightly attached.In this talk I will present recent results about graphs admitting a half-

arc-transitive group action with at most five alternets: If the number ofalternets is at most three, then the graph is necessarily tightly attached, butthere exist graphs with four and graphs with five alternets which are nottightly attached. The exceptional graphs all admit a partition giving therose window graph R6(5, 4) on 12 vertices as a quotient graph in case of fouralternets, and a particular graph on 20 vertices in the case of five alternets.

24


Speaker: Reinhard Laue ([email protected]) Universität BayreuthCoauthors: Michael Braun, Michael Kiermaier, and Axel KohnertTitle: Large sets of q-analogs of designsAbstract: Joining small large sets of t-designs to form large large sets oft-designs allows to recursively construct infinite series of t-designs. Thisconcept is generalized from ordinary designs over sets to designs over finitevector spaces, i.e. designs over GF (q), using three types of joins. While thereare only very few general constructions of such q-designs known so far, fromonly one large set in the literature and two new ones in this paper this waymany infinite series of large sets of q-designs with constant block sizes arederived.

Speaker: Josef Lauri ([email protected]) University of MaltaCoauthors: Russell Mizzi and Raffaele ScapellatoTitle: Two-fold isomorphismsAbstract: Let G and H be two mixed graphs (which can have loops, arcsand edges, that is, self-paired arcs) and let α, β be two bijections from V (G)to V (H) such that (u, v) is an arc in G if and only if (uα, vβ) is an arc in H.Then the pair (α, β) is said to be a two-fold (TF) isomorphism from G toH and, if G = H, a two-fold automorphism. In this talk we shall describehow TF-isomorphisms are related to other problems in graph theory, suchas, stability of graphs and the reconstruction of a graph from its neighbour-hood sets and we shall look at some properties of TF-isomorphisms, such asthe existence of asymmetric graphs with non-trivial TF-automorphisms, andgraphs with TF-rank equal to 1.

Speaker: Weiqiang Li ([email protected]) University of DelawareCoauthor: Sebastian M. CioabaTitle: The extendability of matchings in strongly regular graphsAbstract: A graph G of even order v is called t-extendable if it contains aperfect matching, t < v/2 and any matching of t edges is contained in someperfect matching. The extendability of G is the maximum t such that G ist-extendable. In this talk, we study the extendability properties of stronglyregular graphs. We improve previous results and classify all strongly regulargraphs that are not 3-extendable. We also show that strongly regular graphsof valency k ≥ 3 with λ ≥ 1 are bk/3c-extendable (when µ ≤ k/2) and dk+1

4 e-extendable (when λ > k/2), where λ is the number of common neighbors ofany two adjacent vertices and µ is the number of common neighbors of any

25


two non-adjacent vertices. Our results are close to being best possible asthere are strongly regular graphs of valency k that are not dk/2e-extendable.We show that the extendability of many strongly regular graphs of valencyk is at least dk/2e − 1 and we conjecture that this is true for all stronglyregular graphs of even order.

Speaker: Valery Liskovets ([email protected]) Institute of Mathe-matics, National Academy of Sciences, BelarusTitle: Some recent advances in the analytic enumeration of circulant graphsAbstract: In the last years little progress has been achieved in the enumer-ation of circulant graphs. Here I survey briefly recent results obtained forcirculant graphs of two distinct types: of odd degree and of odd prime powerorder. For the former, in 1998 I formulated a conjecture on isomorphismsof undirected circulant graphs of arbitrary even order 2m, that differ by thepresence of m in their connection sets (“spokes”). In 2012, M. Muzychukproved it easily (in a generalized form) based on his well-known strong char-acterization of isomorphisms of circulant graphs (2004). As a direct corollarythis confirms the following identity c(2m, 2s + 1) = c(2m, 2s), 0 ≤ s < m,that was discovered empirically and verified by exhaustive search for m ≤ 25(B. McKay, 1995), where c(n, r) denotes the number of undirected circulantgraphs of order n and degree r. For prime power orders, J. Lauri and hisstudent Victoria Gatt have recently succeeded in counting circulant graphsand digraphs of orders p3 for small p. In the analytic enumeration they fol-lowed mainly the general scheme elaborated by R. Pöschel and me (2000). Inmy talk I will touch perspectives of obtaining “closed” formulae for circulantpk-graphs with odd prime p and arbitrary k..

Speaker: Alison Gordon Lynch ([email protected]) Universityof WisconsinTitle: Finite-dimensional irreducible modules for an even subalgebra of Uq(sl2)

Abstract: The quantum algebra Uq(sl2) has connections to Q-polyonomialdistance-regular graphs, tridiagonal pairs of linear transformations, the q-tetrahedron algebra, as well as many other combinatorial and algebraic ob-jects. In 2006, Ito, Terwilliger, and Weng gave a presentation for Uq(sl2)in generators x, y, y−1, z, called the equitable presentation, and showed thatxryszt : r, t ∈ N, s ∈ Z is a basis for Uq(sl2). In 2013, Bockting-Conrad andTerwilliger introduced a subalgebra A of Uq(sl2) spanned by the elementsxryszt : r, s, t ∈ N, r + s + t even. In this talk, we give a presentationfor the algebra A and we show that for every d ≥ 1, there exists a uniqueirreducible A-module of dimension d.

26


Speaker: Martin Mačaj ([email protected]) Comenius Uni-versityTitle: On strongly regular graphs attaining the claw boundAbstract: In 1963 Bose proved that any strongly regular graph Γ whose pa-rameters are geometric (with respect) to a partial geometry with parameters(R,K, T ) is necessarily geometric provided thatK > T+R(R−1)(RT+1)/2.In particular, any SRG whose parameters are geometric to the dual geometryof a Steiner Triple System of order n is isomorphic to the block intersectiongraph of some STS(n), provided that n ≥ 69.In 1979 Neumaeir improved Bose’s results by proving that any SRG Γ

with r > 12s(s + 1)(µ + 1)1 (here r > s are the remaining eigenvalues of Γ)

is geometric to some partial geometry whose parameters (R,K, T ) satisfyK > T + R(R − 1)(RT + 1)/2 and T ∈ R − 1, R. Therefore the boundr > 1

2s(s+ 1)(µ+ 1)1 is known as the Neumaier claw bound.We prove that any strongly regular graph with the smallest eigenvalue

s ≤ −3, which attains the claw bound of Neumaier, is isomorphic to the pointgraph of some partial geometry with parameters (R,K, T ) where R = −s,T ∈ R − 1, R and K = T + R(R − 1)(RT + 1)/2. As applications of ourresult we prove the non-existence of SRG’s for two infinite families of feasibleparameter sets and we show that any strongly regular graph with parameters(737, 96, 35, 9) is isomorphic to the block intersection graph of some Steinertriple system of order 67.

Speaker: Alexandr Makhnev ([email protected]) Institute ofMathematics and Mechanics, Rusian Academy of Sciences, UralTitle: Distance-regular extensions of strongly regular graphs with eigen-value 3Abstract: J. Koolen suggested the problem of describing distance-regulargraphs with strongly regular local subgraphs having second eigenvalue atmost t for some positive integer t. A. Makhnev reduced this problem fort = 3 to the case when local subgraphs are exeptional graphs with secondeigenvalue 3. The final result is here stated:Theorem. Let Γ be a distance-regular graph with strongly regular local sub-graphs having second eigenvalue r, 2 < r ≤ 3. If d(Γ) > 2 and u is a vertexof Γ, then one of the following holds:(1) Γ(u) is the union of isolated 4-cliques.(2) Γ(u) is the conference-graph with parameters (4n+ 1, 2n, n− 1, n) (7 ≤n ≤ 12), or a pseudo-geomenric graph for pG3(6, t) and Γ is the Taylor graph.(3) Γ(u) is the 5× 5-grid and Γ is the Johnson graph J(10, 5), or Γ(u) is the

27


pseudo-geomenric graph for GQ(4, 2), GQ(4, 6) and Γ has intersection array45, 32, 12, 1; 1, 6, 32, 45 or 125, 96, 1; 1, 48, 125 respectively.(4) Γ(u) has parameters (21, 10, 5, 4), (96, 19, 2, 4), (99, 14, 1, 2), (115, 18, 1, 3),(169, 42, 5, 12) or (256, 51, 2, 12), d(Γ) = 3 and intersection arrays for Γ areknown.

Speaker: William J. Martin ([email protected]) Worcester PolytechnicInstituteTitle: Graph homotopy, ideals of finite varieties and a surprising dualityAbstract: In 2000, H. Lewis introduced a sequence of subgroups of the ho-motopy group of a graph. These ideas lead to a number of interesting con-jectures relating to the girth of graphs. A seemingly unrelated set of ideasand questions have been studied recently in the context of eigenspace repre-sentations of Q-polynomial association schemes. Let X be a spherical codeor design, considered as a finite variety, and let I = I(X) denote the idealof polynomials that vanish at each point of X. What is the smallest de-gree of a (non-trivial) polynomial in I? What is the smallest k such that Iadmits a generating set of polynomials all having degree k or less? These“dual girth” parameters have interesting connections to properties of spheri-cal designs and association schemes. And there are some natural conjecturesabout these parameters as well. After briefly introducing these two sets ofideas, we focus on a surprising duality between the two which occurs in thecase of abelian groups.

Speaker: Dragan Marušič ( [email protected]) University of Pri-morskaTitle: Semiregular subgroups of transitive permutation groupsAbstract: In this talk I will discuss the semiregularity problem which inits original form asks if every vertex-transitive digraph has a nonidentitysemiregular automorphism, that is an automorphism with all orbits of thesame size. I will give a short overview of the current status of this problem(and some of its generalizations) and discuss possible future directions.

Speaker: Karen Meagher ([email protected]) University ofReginaTitle: The Erdős-Ko-Rado Theorem: an algebraic perspectiveAbstract: The Erdős-Ko-Rado Theorem is a major result in extremal settheory. It gives the exact size and structure of the largest system of sets,with a fixed number of elements, that has the property that any two sets

28


in the system have non-trivial intersection. There are many extensions ofthis theorem to combinatorial objects other than set systems, such as vectorssubspaces over a finite field, integer sequences, partitions, and recently, therehave been several results that extend the EKR theorem to permutations.I will describe an algebraic method that can be used to prove the EKR

theorem for several types of combinatorial objects. My focus in this talk willbe permutations. I will use this method to show how the natural extensionof the EKR Theorem holds for two permutation groups. This method relieson having knowledge of the irreducible representations of the group and maywell be applicable to other groups whose irreducible representations are wellunderstood.

Speaker: Mike Newman ([email protected]) University of OttawaCoauthor: Amin BahmanianTitle: Embedding factorizations of hypergraphsAbstract: We consider the question of embedding factorizations in hyper-graphs. Given an r-factorization of the complete h-uniform hypergraph Kh

m,we ask when can it be embedded in a factorization of Kh

n? There are some“obvious” necessary conditions; we consider the question of when they aresufficient as well. We are partly motivated by a paper of Häggkvist andHellgren which answers the question in the special case of r = 1. We usethe amalgamation-detachment technique to reduce the hypergraph problemto a problem on multisets of integers, and a group action on a finite set to(mostly) solve this for feasible r, as well as further generalizations.

Speaker: Dmitrii V. Pasechnik ([email protected]) Uni-versity of OxfordCoauthors: Swee Hong Chan and Henk D.L. HollmannTitle: Sandpile groups of generalized de Bruijn and Kautz graphs and circu-lant matrices over finite fieldsAbstract: A maximal minor M of the Laplacian of an n-vertex Euleriandigraph Γ gives rise to a finite group Zn−1/Zn−1M known as the sand-pile (or critical) group S(Γ) of Γ. We determine S(Γ) of the generalized deBruijn graphs Γ = B(n, d) with vertices 0, 1, . . . , n − 1 and arcs (i, di + k)for 0 ≤ i ≤ n − 1 and 0 ≤ k ≤ d − 1, and closely related generalized Kautzgraphs, extending and completing earlier results for the classical de Bruijnand Kautz graphs.Moreover, for a prime p and an n-cycle permutation matrix X ∈ GLn(p)

we show that S(DB(n, p)) is isomorphic to the quotient by 〈X〉 of the cen-

29


traliser of X in PGLn(p). This offers an explanation of the coincidence ofnumerical data in the sequences A027362 and A003473 of the OEIS, and al-lows one to speculate upon the possibility of constructing normal bases in thefinite field Fnp from spanning trees in DB(n, p). Preprint: arXiv:1405.0113.

Speaker: Christian Pech ([email protected]) TechnischeUniversität DresdenTitle: On highly regular strongly regular graphsAbstract: This talk will be about the t-vertex condition and other relatedregularity-conditions for strongly regular graphs. Such conditions are usedas combinatorial approximations of homogeneous graphs. Graphs that fulfillthe t-vertex condition for t ≥ 4 turn out to be extremely rare. This led M.Klin to conjecture that there might be t0 such that strongly regular graphsthat fulfill the t-vertex condition for t ≥ t0, are in fact 2-homogeneous (i.e.they have an automorphism group that acts transitively on vertices, arcs,and non-arcs).In this talk I am going to report about new results on the t-vertex condition

of the point graphs of partial linear spaces, strengthening results by S. Re-ichard. In particular, we show that the point graphs of partial quadranglesof order (q − 1, q2, q2 − q) satisfy the 6-vertex condition and that the pointgraphs of generalized quadrangles of order (q, q2) are (3, 7)-regular (this is aregularity condition strictly stronger than the 7-vertex condition).This is part of a more wide project of analyzing regularity conditions of

strongly regular graphs in order to prove or to refute Klin’s conjecture.

Speaker: Ilia Ponomarenko ([email protected]) Steklov Institute ofMathematics, Russian Academy of Sciences, St. PetersburgCoauthor: Sergei EvdokimovTitle: Coset closure of a circulant S-ring and the schurity problemAbstract: Let G be a finite group. There is a natural Galois correspondencebetween the permutation groups containing G as a regular subgroup, andthe Schur rings (S-rings) over G. The problem I deal with in my talk, is tocharacterize those S-rings that are closed under this correspondence, whenthe group G is cyclic (the schurity problem for circulant S-rings). Recently,Sergei Evdokimov and the speaker proved that up to a natural reduction,the characteristic property of such an S-ring is to be a certain algebraic fu-sion of its coset closure. In my talk I will explain this result and some of itscorollaries. In particular, I will show that the schurity problem is equivalentto the consistency of a modular linear system associated with a circulantS-ring under consideration.

30


Speaker: Reinhard Pöschel ([email protected]) Tech-nische Universität DresdenCoauthor: Mikhail KlinTitle: Automorphism groups of Schur rings over cyclic groups of prime-powerorderAbstract: Schur rings are a well-known tool for the investigation of permu-tation groups and associated algebraic and combinatorial structures (e.g.,association schemes, coherent configurations). The automorphism groupsof Schur rings are of special interest because they describe “symmetries” ofcombinatorial Cayley objects, in particular automorphism groups of circulantgraphs, i.e., Cayley graphs over a cyclic group Zn. In this talk we give someinsight into the long history of determining the automorphism group of Schurrings, in particular for Schur rings over the cyclic groups Zpm of prime-powerorder. We describe so-called subwreath products of permutation groups andshow how this approach can lead to an explicit and constructive presentationof the automorphism group of an arbitrary Schur ring over Zpm .

Speaker: Sven Reichard ([email protected]) Technische Uni-versität DresdenTitle: Three-class association schemes related to a construction by MathonAbstract: In a classical paper, R. Mathon gave a cyclotomic construction ofassociation schemes in order to improve lower bounds for Ramsey numbers.Many of the basis graphs are antipodal distance regular graphs (drg’s) ofdiameter 3, hence they yield three-class schemes as mergings. Most of themare non-Schurian. We give a proper generalization of his construction, showthat a cyclic group acts on the drg’s, and use difference sets in that groupin order to get new three-class mergings. Those are imprimitive associationschemes with new parameters on (q2− 1)/m points for a prime power q andan even divisor m of q − 1. In this context Mathon’s orignal constructioncorresponds to trivial difference sets, i.e., singletons. Extensive computerexperiments suggest that our new schemes are non-Schurian. We concludewith a discussion of our attempts to prove this conjecture.

Speaker: Nitin Saxena ([email protected]) Indian Institute of Technol-ogy, KanpurCoauthors: Manuel Arora, Gábor Ivanyos, and Marek KarpinskiTitle: Combinatorial schemes in algebraic algorithmsAbstract: We study a recently developed combinatorial object called m-scheme. It captures several standard algebraic objects like permutation

31


groups, strongly regular graphs and association schemes. An m-scheme onpoints V is, basically, a partition of V m satisfying certain natural conditions.We work with a conjecture about schemes, especially about their intersectionnumbers, and prove it in special cases. Finally, we show how it applies to theclassical computational problem of factoring polynomials over finite fields.Our proof techniques relate to association schemes and their high dimensiongeneralizations.

Speaker: Jonathan D. H. Smith ([email protected]) Iowa StateUniversityCoauthors: Pranava K. Jha and Giora SlutzkiTitle: Circulant graphs, nonlinear loop transversal codes and nonassociativeloopsAbstract: Loop transversal (error-correcting) codes are constructed by plac-ing loop or quasigroup structure on the set of errors to be corrected bythe code. Applications of the method have hitherto concentrated on linearcodes. Although the method has been very effective in designing codes, andhas proved especially useful in handling unconventional error patterns, thequasigroup structure always reduces to an abelian group in the linear con-text.

The current talk reports on recent work applying the loop-transversalmethod to the open problem of perfect code construction and classifica-tion in circulant graphs. Most notably, it is shown how nonlinear single-error-correcting loop-transversal codes (efficient dominating sets) in circu-lant graphs may lead to nonassociative loop structure on the balls that formthe error sets.

Speaker: Leonard Soicher ([email protected]) Queen Mary Uni-versity of LondonTitle: On cliques in edge-regular graphsAbstract: A graph Γ is edge-regular with parameters (v, k, λ) if Γ has ex-actly v vertices, is regular of degree k, and every pair of adjacent verticeshave exactly λ common neighbours. We show how to apply a certain “blockintersection polynomial” in two variables to determine a good upper boundon the clique number of an edge-regular graph Γ, given only its parameters,and then the application of this polynomial to obtain results on the cliques Sof Γ with the property that every vertex of Γ not in S is adjacent to exactlym or m+ 1 vertices of S, for some constnt m ≥ 0.

32


Speaker: Sung Y. Song ([email protected]) Iowa State UniversityCoauthor: Katy NowakTitle: Cyclotomic association schemes and their fusion and fission schemesAbstract: Motivated by the recent work of Feng, Momihara, Wu and Xiang,we investigate the characteristics of many association schemes that are fusionschemes and fission schemes of cyclotomic association schemes. In this talk,we discuss various constructions of association schemes of small class (classup to twelve) that directly arise from cyclotomy of finite fields. We thendescribe their fusion and fission relations. We also discuss the properties ofthe relation graphs of these association schemes if time permits.

Speaker: Michael Tait ([email protected]), University of Californiaat San DiegoCoauthor: Craig TimmonsTitle: Orthogonal polarity graphs and Sidon SetsAbstract: Let q be a prime power and let θ generate Fq2 . The set

A(q, θ) := aZq2−1 : θa − θ ∈ Fq

is called a Bose-Chowla Sidon set. In this talk, we use the Cayley sum graphconstructed using A(q, θ) to construct the Erdős-Rényi Polarity graph in anew way. We also discuss applications to the Turán number ex(n,C4).

Speaker: Paul Terwilliger ([email protected]) University of Wis-consinTitle: Billiard Arrays and finite-dimensional irreducible Uq(sl2)-modulesAbstract: In this talk we will describe the notion of a Billiard Array. This isa triangular array of one-dimensional subspaces of a finite-dimensional vectorspace, subject to several conditions that specify which sums are direct. Weuse Billiard Arrays to characterize the finite-dimensional irreducible Uq(sl2)-modules, for q not a root of unity. The equitable presentation of Uq(sl2)comes up naturally in this context.

Speaker: Katarína Tureková ([email protected]) Comenius Uni-versityCoauthor: Martin MačajTitle: Graphs similar to strongly regular graphsAbstract: The degree/diameter problem is to find the largest possible graphwith given diameter d and given maximum degree k. For graphs with di-

33


ameter 2, the upper bound (Moore bound) simplifies to k2 + 1. In 1980Erdős, Fajtlowicz and Hoffman showed that, with the exception of the cycleof length 4, there does not exist a k-regular graph with diameter 2 and k2

vertices (one less vertex than in the Moore bound). They reduced this prob-lem to solving the matrix equation A2 + A− (k − 1)I = J +K, where A isthe adjacency matrix of the graph, I is the identity matrix, J is the all-onesmatrix, and K is the matrix of a suitable 1-factor.Our aim is to generalise this problem by replacing Moore graphs by strongly

regular graphs. That is, we are looking for k-regular graphs on n verticessuch that their adjacency matrix A satisfies the equation

A2 + (c− a)A+ (c− k)I = cJ +K.

We derive necessary conditions for parameters (n, k, a, c) analogous to theintegral criterion for strongly regular graphs. Here, systemic application ofalgebraic properties of A3 proves to be crucial. Finally we find the complete(infinite) list of parameters satisfying these necessary conditions. Existenceof graphs with these parameters remains an open problem.

Speaker: Vasyl Ustimenko ([email protected]) University ofMaria Curie SkłodowskaCoauthor: Urszula Romańczuk-PolubiecTitle: On algebraic constructions of graphs without small cycles and com-mutative diagrams and their applicationsAbstract: We call Γ a balanced directed graph if it is a directed graph with-out multiple arrows and the numbers of inputs and outputs are the same forevery vertex.The concepts of regular graphs, small worlds graphs, graphs of large girth

and graphs with large cycle indicator can be naturally generalized for theclass of balanced directed graphs.We prove, that for each pair (K,S), where K is commutative ring and

S be its multiplicatively closed subset without zero, there exists an infinitedirected regular balanced graph ΓS(K) without commutative diagrams. Wewill use the well defined functor (K,S) → ΓS(K) for the construction offamilies of graphs of large girth, graphs with large cycle indicator, and smallworld graphs for which ΓS(K) will appear as a well defined projective limit.A brief survey of applications of ΓS(K) to Information Security will be given.

34


Speaker: Andrey Vasil’ev ([email protected]) Sobolev Institute ofMathematics, Russian Academy of Sciences, NovosibirskCoauthor: Ilya PonomarenkoTitle: On non-abelian Schur groupsAbstract: Let G be a finite group. Following Wielandt a subring A of thering ZG is called a Schur ring or S-ring over the group G if there exists apartition S of this group such that

e ∈ S and S−1 = S (1)

where S−1 = X−1 : X ∈ S with X−1 = x−1 : x ∈ G; and

A = SpanX : X ∈ S (2)

where X is the sum of elements of X in the group ring.If Γ is a permutation group with Gright ≤ Γ ≤ Sym(G) and S is the set

of orbits of the stabilizer of the identity e = eG in Γ, then Z-submoduleA = A(Γ, G) = SpanX : X ∈ S of the group ring ZG is an S-ring asit was observed by Schur. Following Pöschel an S-ring A over G is saidto be schurian if there exists a suitable permutation group Γ such thatA = A(Γ, G).A finite group G is called a Schur group if every S-ring over G is schurian.

We prove that every non-abelian Schur group G is metabelian and the num-ber of distinct prime divisors of the order of G is at most 7.

Speaker: Jason Vermette ([email protected]) University ofDelawareCoauthors: Sebastian Cioaba and Willem HaemersTitle: Spectral properties of simplicial rook graphsAbstract: The simplicial rook graph is the graph whose vertices are latticepoints in the nth dilate of the standard simplex in Rd, with two vertices ad-jacent if and only if they differ in exactly two coordinates. Seen another way,they are the graphs whose vertices are weak compositions of n with d partswith the same adjacency relation. Martin and Wagner found the spectrumof SR(3, n), which is integral, and conjectured that SR(d, n) is integral forany d and n. We use a vertex partition and quotient matrix to find a partialspectrum of SR(d, n) consisting only of integers. We also find the completespectrum of SR(d, 3), which is integral, and construct nonisomorphic graphscospectral with SR(4, n) and SR(d, 3) using Godsil-McKay switching. Inthis talk I will discuss these and other spectral properties of the simplicialrook graphs.

35


Speaker: Andrew P Wang ([email protected]) Northern Illi-nois UniversityTitle: Coxeter table algebrasAbstract: Zieschang’s Coxeter schemes give an alternative treatment of Tits’theory of spherical buildings. We abstract the adjacency algebras of Coxeterschemes to integral table algebras that we call Coxeter table algebras, andshow that these algebras must be generic algebras of Bourbaki and Couil-lens. In the case where a Coxeter table algebra has a generating set of thickinvolutions, we show that Zieschang’s exchange condition can be replacedwith a much weaker hypothesis.

Speaker: Ye Wang ([email protected]) Tongji UniversityCoauthor: Felix LazebnikTitle: On some cycles in Wenger graphsAbstract: Let p be a prime, q be a power of p, and let Fq be the field of qelements. For any positive integer n, the Wenger graph Wn(q) is defined asfollows: it is a bipartite graph with the vertex partitions being two copies ofthe (n+1)-dimensional vector space Fn+1

q , and two vertices p = (p1, . . . , pn+1)

and l = [l1, . . . , ln+1] being adjacent if pi+ li = p1li−11 , for i = 2, 3, . . . , n+ 1.

In 2008, J.Y. Shao, C.X. He and H.Y. Shan showed that for n ≥ 2, Wn(q)contains a cycle of length 2k where 4 ≤ k ≤ 2p and k 6= 5. In our paper, weextend their results by showing that(i) W1(q) contains cycles of length 2k, where k = 3, 4, . . . , q2 − q + 1,(ii) For n ≥ 2,Wn(q) contains cycles of length 2k, where k = 4, 6, 7, . . . , 4q+1and k = p2 − p,(iii) For n ≥ 2,Wn(q) contains cycles of length at least bcqαc, where c = 2.37and α = 1.58.

Speaker: Jason Williford ([email protected]) University of WyomingTitle: Q-polynomial association schemesAbstract: An association scheme can be thought of as a combinatorial gen-eralization of the orbitals of a finite permutation group. In the 1973 the-sis of Philippe Delsarte, the author identified two special classes of asso-ciation schemes: the so-called P -polynomial and Q-polynomial schemes.The schemes which are P -polynomial are precisely those generated by adistance-regular graph. However, the Q-polynomial schemes have no analo-gous combinatorial definition. In this talk, we will discuss known examplesof Q-polynomial schemes, including a recently discovered family with an un-

36


bounded number of classes. Emphasis will be on connections with finitegeometry and designs, with open problems.

Speaker: John Wilmes ([email protected]) University ofChicagoCoauthor: Xiaorui SunTitle: Geometric clique structures in association schemesA coherent configuration is a directed generalization of an association scheme.An old conjecture of Babai states that for every ε > 0, every sufficiently largeprimitive coherent configuration with ≥ exp(nε) automorphisms has a primi-tive automorphism group. No progress has been made on the conjecture sinceBabai’s 1981 paper in the Annals of Math. established it for all ε > 1/2. Thepresent work establishes the conjecture for all ε > 1/3 assuming boundedrank.A key step in our work is proving the existence of geometric clique struc-

tures in constituent graphs of primitive coherent configurations in a certainrange of the parameters. Specifically, given certain inequalities on the pa-rameters of primitive coherent configuration, we find a union of constituentgraphs with a collection of cliques of asymptotically equal order ≥ 3 withthe property that every edge lies in a unique clique. Such clique structuresappear in many familiar graphs, including Johnson graphs, Hamming graphsover non-binary alphabets, and more generally line-graphs of partial geome-tries.

Speaker: Bangteng Xu ([email protected]), Eastern Kentucky Uni-versityTitle: Multiplicities of irreducible characters of table algebras and applica-tions to association schemesAbstract: Commutative standard table algebras with exactly one multiplic-ity not equal to 1 are characterized by the wreath product of some specialtable algebras by Antonou. In this talk we present properties of table alge-bras with fused centers. Then we give characterizations of arbitrary tablealgebras which have exactly one irreducible character whose multiplicity anddegree are not equal in terms of table algebras with fused centers and thewreath product of their fused centers. Applications to association schemesare also discussed.

37


Speaker: Shenglin Zhou ([email protected]) South China Universityof TechnologyCoauthors: Haiyan Guan and Yan ZhuTitle: Flag-transitive symmetric designs with (r, λ) = 1 and alternating socleAbstract: A (v, k, λ)-symmetric design is an incidence structure consistingof a set P of v points, a set B of blocks (which can be thought of as subsetsof P ) and an incidence relation such that: i) |P | = |B|, ii) Every block isincident with exactly k points, and iii) Every pair of points is incident withexactly λ blocks. An automorphism of a symmetric design is a permutationof the points which also permutes the blocks. A flag in a symmetric designis an incident point-block pair. An automorphism group G acting on asymmetric design D is flag-transitive if it acts transitively on the set of flagsof D. In this talk, we will discuss the classification of flag-transitive (v, k, λ)-symmetric designs D with (r, λ) = 1 and Soc(G) = An, for n ≥ 5. We provethat D is the projective space PG2(3, 2) and G = A7.

Speaker: Paul-Hermann Zieschang ([email protected])University of Texas at BrownsvilleTitle: Hypergroups, association schemes, buildingsAbstract: Let S be a set, and let µ be a map from S × S to the power setof S. For any two elements p and q of S, we write pq instead of µ(p, q) andassume that pq is not empty. For any two non-empty subsets P and Q of S,we define the complex product PQ to be the union of the sets pq with p ∈ Pand q ∈ Q. If one of the two factors in a complex product consists of a singleelement, say s, we write s instead of s in that product.Following (and generalizing) Frederic Marty’s terminology (1934) we call

S a hypergroup (with respect to µ) if the following three conditions hold.1. ∀ p, q, r ∈ S: p(qr) = (pq)r.2. ∃ e ∈ S ∀ s ∈ S: se = s.3. ∀ s ∈ S ∃ s∗ ∈ S ∀ p, q, r ∈ S: p ∈ q∗r∗ ⇒ q ∈ r∗p∗ and r ∈ p∗q∗.

Each association scheme satisfies the above three conditions with respectto its complex multiplication. Thus, hypergroups generalize associationschemes.I will explain how association schemes may take advantage of the structure

theory of hypergroups. Special attention will be given to the embedding ofthe theory of buildings and twin buildings into scheme theory and to factor-ization of schemes of finite order.

38


Speaker: Matan Ziv-Av ([email protected]) Ben-Gurion University ofthe NegevTitle: Constructive enumeration of coherent configurations of small orderAbstract: Hanaki and Miyamoto, combining theoretical reasoning and com-puter search, classified association schemes of order up to 30, as well as order32, 33, 34 and 38. One of the results of the classification is that the smallestnon-Schurian scheme is of order 15. Classification of coherent configurationis harder. First attempts in this direction (Nagatomo and Shigezumi, Re-ichard) did not agree on results. We use the classification of associationschemes to enumerate coherent configurations of small order. The first tar-get of classification is to find the smallest order of a non-Schurian coherentconfiguration. We discuss the methods and challenges of this ongoing task,paying attention to verification of results.

39


List of Participants

Aida AbiadTilburg University, [email protected]

Angela AntonouNorthern Illinois University, [email protected]

László BabaiUniversity of Chicago, [email protected]

Rosemary BaileyUniversity of St. Andrews, [email protected]

Alexander BargUniversity of Maryland, [email protected]

Taylor BerrangVillanova University, [email protected]

Anton BettenColorado State University, [email protected]

Sarah Bockting-ConradUniversity of Wisconsin, [email protected]

Peter CameronUniversity of St. Andrews, [email protected]

40


Carrie CaswellVillanova University, [email protected]

Patrick CesarzUniversity of Delaware, [email protected]

David ChandlerWidener University, [email protected]

Sebastian CioabaUniversity of Delaware, [email protected]

Brian CurtinUniversity of South Florida, [email protected]

Edwin van DamTilburg University, [email protected]

Aiping DengDonghua University, [email protected]

Matt DevosSimon Fraser University, [email protected]

Ted DobsonMississippi State University, [email protected]

Yun FanCentral China Normal University, [email protected]

41


Igor FaradjevISA RAS, Moscow, [email protected]

Frank FiedlerWesley College, [email protected]

Miquel Angel FiolUniversitat Politecnica de Catalunya, [email protected]

Alexander GavrilyukTohoku University, [email protected]

Gary GreavesTohoku University, [email protected]

Victor GrinbergPittsburgh, Pennsylvania, [email protected]

Štefan GyürkiSlovak University of Technology, Slovak [email protected]

Willem HaemersTilburg University, [email protected]

Akihide HanakiShinshu University, [email protected]

Kathryn HaymakerUniversity of Nebraska, [email protected]

42


Paul HeardingUniversity of Delaware, [email protected]

Allen HermanUniversity of Regina, [email protected]

Mitsugu HirasakaPusan National University, South [email protected]

Katarína HriňákováSlovak University of Technology, Slovak [email protected]

Alexander HulpkeColorado State University, [email protected]

Alexander IvanovImperial College, [email protected]

Kenneth JohnsonPennsylvania State University at Abington, [email protected]

Gareth JonesUniversity of Southampton, [email protected]

Leif JørgensenAalborg University, [email protected]

Aleksandar JurišičUniversity of Ljubljana, [email protected]

43


Grigory KabatianskyIITP RAS, Moscow, [email protected]

Ján KarabášMatej Bel University, Slovak [email protected]

Anna KasikovaBowling Green State University, [email protected]

Alexander KelmansUPR, Rutgers University, Puerto [email protected]

Caroline KettlestringsNorthern Illinois University, [email protected]

Douglas KleinTexas A & M University, Galveston, [email protected]

Mikhail KlinBen-Gurion University of the Negev, [email protected]

Aleksandr KodessUniversity of Delaware, [email protected]

Elena KonstantinovaSobolev Institute RAS, Novosibirsk, [email protected]

Jack KoolenUniversity of Science and Technology of China, [email protected]

44


Kristína KováčikováComenius University, Slovak [email protected]

István KovácsUniversity of Primorska, [email protected]

Brian KronenthalKutztown University, [email protected]

Klavdija KutnarUniversity of Primorska, [email protected]

Reinhard LaueUniversity of Bayreuth, [email protected]

Josef LauriUniversity of Malta, [email protected]

Felix LazebnikUniversity of Delaware, [email protected]

Weiqiang LiUniversity of Delaware, [email protected]

Valery LiskovetsInstitute of Mathematics, NAS, [email protected]

Alison Gordon LynchUniversity of Wisconsin, [email protected]

45


Martin MačajComenius University, Slovak [email protected]

Alexandr MakhnevInstitute of Math. & Mechanics UB, [email protected]

William MartinWorcester Polytechnic Institute, [email protected]

Dragan MarušičUniversity of Primorska, [email protected]

Karen MeagherUniversity of Regina, [email protected]

Eric MoorhouseUniversity of Wyoming, [email protected]

Mikhail MuzychukNetanya Academic College, [email protected]

Roman NedelaInstitute of Mathematics, Slovak Acad. Sci., Slovak [email protected]

Mike NewmanUniversity of Ottawa, [email protected]

Dmitrii PasechnikUniversity of Oxford, [email protected]

46


Christian PechTechnische Universität, Dresden, [email protected]

Rafael PlazaUniversity of Delaware, [email protected]

Ilia PonomarenkoSt. Petersburg Mathematical Institute, [email protected]

Reinhard PöschelTechnische Universität, Dresden, [email protected]

Sven ReichardTechnische Universität, Dresden, [email protected]

Joseph ReiterVillanova University, [email protected]

Carolyn RomanoVillanova University, [email protected]

Alyssa SankeyUniversity of New Brunswick, [email protected]

Nitin SaxenaIndian Institute of Technology, [email protected]

Sergey ShpectorovUniversity of Birmingham, [email protected]

47


Jana ŠiagiováComenius University, Slovak [email protected]

Jozef SiranSlovak University of Technology, Slovak [email protected]

Martin ŠkovieraComenius University, Slovak [email protected]

Jonathan SmithIowa State University, [email protected]

Leonard SoicherQueen Mary, University of London, [email protected]

Sung-Yell SongIowa State University, [email protected]

Pablo SpigaUniversity of Milan, [email protected]

Shuying SunUniversity of Delaware, [email protected]

Michael TaitUniversity of California at San Diego, [email protected]

Itzahak TamoUniversity of Maryland, [email protected]

48


Paul TerwilligerUniversity of Wisconsin, [email protected]

Katarína TurekováComenius University, Slovak [email protected]

Vasyl UstimenkoUniversity of Maria Curie-Sklodowska, [email protected]

Andrey Vasil’evSobolev Institute of Mathematics, Novosibirsk, [email protected]

Jason VermetteUniversity of Delaware, [email protected]

Andrew WangNorthern Illinois University, [email protected]

Ye WangTongji University, [email protected]

Jason WillifordUniversity of Wyoming, [email protected]

John WilmesUniversity of Chicago, [email protected]

Andrew WoldarVillanova University, [email protected]

49


Qing XiangUniversity of Delaware, [email protected]

Bangteng XuEastern Kentucky University, [email protected]

Shenglin ZhouSouth China University of Technology, [email protected]

Paul-Hermann ZieschangUniversity of Texas at Brownsville, [email protected]

Matan Ziv-AvBen-Gurion University of the Negev, [email protected]

50

Modern Trends in Algebraic Graph Theory...(di)graph Gin terms of jVj, r, and the Laplacian...

Documents

Transcript of Modern Trends in Algebraic Graph Theory...(di)graph Gin terms of jVj, r, and the Laplacian...