Thesis exam Regular t-graphs, antipodal distance regular ...dannykal/research/thesis exam...

Two-graphs and regular two-graphst-graphs and regular t-graphs

Regular 3-graphsNew constructions

Thesis examRegular t-graphs, antipodal distance regular graphsof diameter 3 and related combinatorial structures

Daniel Kalmanovich

Ben Gurion University of the Negev

May 29, 2011

Daniel Kalmanovich Regular 3-graphs and ADRG’s of diameter 3



Layout

1 Two-graphs and regular two-graphs

2 t-graphs and regular t-graphs

3 Regular 3-graphs

4 New constructions




Two-graphs

Definition

Let X be a set of n elements called vertices. X {3} is the set of all3-subsets of X . A subset ∆ ⊆ X {3} is a two-graph if every 4-subsetof X contains an even (∈ {0, 2, 4}) number of members of ∆.Sometimes we call (X ,∆) a two-graph, and ∆ the set of oddtriples.




Cohomological definition

Definition

A two-graph (X ,∆) is a function:

f : X {3} −→ U2

such that f (x) = −1⇐⇒ x ∈ ∆, satisfying:

f ({x , y , z}) · f ({x , y , t}) · f ({x , z , t}) · f ({y , z , t}) = 1

for any {x , y , z , t} ∈ X {4}.




The (Seidel) adjacency matrix

Definition

The (Seidel) adjacency matrix A = (ai ,j) of a graph Γ = (V ,E ) isa {0,−1, 1}-matrix having:

ai ,j =

0 i = j−1 {i , j} ∈ E1 {i , j} /∈ E




Switching in matrix terms

In this notation, if the graph Γ′ is obtained from Γ by switchingwith respect to X ⊆ V , then its adjacency matrix A′ is obtainedfrom A by a similarity transformation by a diagonal matrix having{−1, 1} on its diagonal. Explicitly:

A′ = DAD

Where Di ,i = −1⇐⇒ i ∈ X .




Taylor and Levingston showed that there is a one-to-onecorrespondence between two-graphs and antipodal 2-fold covers ofcomplete graphs. In this interpretation, switching corresponds tocyclic permutations of the fibers, and thus non-isomorphic graphscorrespond to different two-graphs. Such a cover, when it is alsodistance regular is called a Taylor graph, these are distance regulargraphs with intersection array

{k , µ, 1; 1, µ, k} .




Equivalence of Taylor graphs and regular two-graphs

Consider the combinatorial definition of a regular two-graph:

Definition

A two-graph (X ,∆) is called regular if every 2-subset{x , y} ∈ X {2} is contained in the same number of triples from ∆.

In the correspondence established by Taylor and Levingston, Taylorgraphs correspond to regular two-graphs. This will be a particularcase of our more general result to come.




Layout



3 Regular 3-graphs

4 New constructions




Cohomological definition

A few preliminary definitions and notations:

A p-cochain into Ut is a function f : X p −→ Ut such that:

f (x) = 1 for every x ∈ X p in which xi = xj for some i 6= j .f (x) = σ(f (y)) whenever y is the result interchanging xi andxj for some i 6= j .

The coboundary operator δ is a function from the set ofp-cochains to the set of (p + 1)-cochains defined by:

δf (x1, . . . xp+1) =

p+1∏i=1

σi (f (x̂i ))

where σ is the inverse operation in Ut , andx̂i = (x1, . . . , xi−1, xi+1, . . . , xp+1).




2-cochains are called weights (on X ).

A p-cochain with vanishing coboundary (this means thatδf = 1) is called a p-cocycle.

Every p-cocycle is the coboundary of a (p − 1)-cochain.

Definition

A t-graph is a 3-cocycle into Ut .




Regular t-graphs

Definition

A t-graph is called regular if for every pair x , y ∈ X , the number ofz ∈ X \ {x , y} such that f (x , y , z) = α is constant. This numberis denoted m(α).

Remark

It is routine to check that in case t = 2, the above definitions areexactly those of two-graphs and regular two-graphscorrespondingly.

For regular 3-graphs, we have:

m(ζ) = m(ζ2).

Denote a = m(1), b = m(ζ) = m(ζ2), then: n = a + 2b + 2.




Interpretation as a t-fold cover of Kn

Let Φ be a t-graph. Let w : X 2 −→ Ut be a weight on X withvalues in Ut such that Φ = δw . Define the graph Γw = (V ,E ),where:

V = X × {1, 2, . . . , t}. For convenience, we denote the vertex(x , i) by xi .

{xi , yj} ∈ E ⇐⇒ w(x , y) = ζ j−i

The resulting graph is a t-fold cover of the complete graph Kn

(n = |X |), and if w(x , y) = ζ i then the set of edges betweenx1, x2, ..., xt and y1, y2, ..., yt form a perfect matching which isgiven by the i th power of the permutation matrix of (1, 2, ..., t).




Layout



3 Regular 3-graphs

4 New constructions




In case t = 3, the corresponding graph Γw is a 3-fold cover of Kn

with exactly 3 types of matchings between fibers:

y

• y3

• y2

• y1

x

•x3

•x2

•x1

w(x , y) = 1

y

• y3

• y2

• y1

x

•x3

•x2

•x1

w(x , y) = ζ

OOOOOOO

OOOOOOO��

y

• y3

• y2

• y1

x

•x3

•x2

•x1

w(x , y) = ζ2

????????? ooooooo ooooooo

Matchings between fibers of Γw




Regular 3-graphs are antipodal distance regular covers

Consider the (combinatorial) definition of a regular 3-graph Φ.Now consider the graph Γw defined by some weight w in theswitching class of Φ (δw = Φ). We have the following:

Proposition

Γw is an antipodal distance regular cover of Kn with parameters(n, 3, b).

Remark

Not every ADRG gives a regular 3-graph as was the case withregular two-graphs.




ADRG’s with cyclic matchings

Definition

Let Γ = (V ,E ) be a labeled (n, r , c2)-cover, i.e. such thatV = X × {1, 2, . . . , r}. Let σ = (1, 2, . . . , r) ∈ Sr . A cyclicmatching in Γ is a matching that corresponds to σi , for some1 ≤ i ≤ r .

Definition

A (not labeled) (n, r , c2)-cover Γ = (V ,E ) is said to be with cyclicmatchings if there exists a labeling of V such that:

all the matchings of Γ are cyclic, and

every cyclic matching appears.




Remark

Note that in an (n, 3, c2)-cover with cyclic matchings, an edgebetween two fibers x , y ∈ X of Γ determines the matching betweenthem. Explicitly, if {xi , yj} ∈ E then the only cyclic matching thatsatisfies this must also satisfy{xi−1 (mod 3), yj−1 (mod 3)}, {xi+1 (mod 3), yj+1 (mod 3)} ∈ E .




We prove:

Proposition

Every regular 3-graph with parameters (n, a, b) defines an(n, 3, b)-cover with cyclic matchings.

And the converse:

Proposition

Every (n, 3, c2)-cover with cyclic matchings defines a regular3-graph with parameters (n, a1, c2).

To prove this proposition, we formulate a ”3 fibers configurationlemma”, which characterizes the induced subgraphs on 3 fibers ina cover with cyclic matchings.




Lemma

Let Γ = (V ,E ) be an (n, 3, c2)-cover with cyclic matchings, and letX be the set of fibers of Γ. Then for any x , y , z ∈ X we have:

(1) Γ|{x ,y ,z} ∼= 3 ◦ C3 or Γ|{x ,y ,z} ∼= C9,

(2) For any two fibers x , y ∈ X the number of z ∈ X such thatΓ|{x ,y ,z} ∼= 3 ◦ C3 is a1,

(3) For any two fibers x , y ∈ X the number of z ∈ X such thatΓ|{x ,y ,z} ∼= C9 is 2c2.




Elements from the Godsil-Hensel paper

Lemma (Lemma 3.1)

An r-fold cover of Kn is antipodal distance regular if and only ifthere exists a constant c2 such that any two non-adjacent verticesfrom different fibers of the cover have exactly c2 commonneighbors.

Definition

The voltage group of Γ is the subgroup T ≤ Aut (Γ) whichstabilizes each fiber of Γ. When T is cyclic of order r (the size of afiber), the graph Γ is called a cyclic cover.




To complete the picture, we use Lemma 7.2 of Godsil and Henselto prove:

Proposition

The cyclic matchings property is equivalent to being a cyclic cover.

Proof.

From the cyclic matchings property we have 〈f 〉 = C3 = 〈(1, 2, 3)〉.By Lemma 7.2, the voltage group

T ∼= CS3(C3) = C3.




By Theorem 9.2 of Godsil and Hensel, we obtain a furtherrestriction on the feasible parameters of regular 3-graphs:

Corollary

If (n, a, b) are the parameters of a regular 3-graph then 3 | n.




Higman’s theory of A4(Γw ) and A6(Γw )

Let Γ := Γw = (V ,E ) be the graph defined by w .

Construction

Define symmetric binary relations {Si}3i=0 partitioning V 2 asfollows:

S0 = diagV 2,

S1 = {(xi , xj) | i 6= j , x ∈ X} ,S2 = E ,

S3 = V 2 \ (S0 ∪ S1 ∪ S2).

Proposition (Higman)

A4 (Γ) :=(V , {Si}3i=0

)is a symmetric (and thus commutative)

association scheme.

Remark

Notice that A4(Γ) with the ordering (S0, S2,S3,S1) is the metricassociation scheme of the ADRG Γ. This is in fact equivalent to Γbeing distance regular, since A4(Γ) is just the distance partition ofΓ (not in the correct ordering).




The association scheme A4 (Γ) admits a coherent rank 6refinement:

Construction

Define:

R0 = diagV 2

R1 ={(

xi , xi+1 (mod 3)

) ∣∣ i = 1, 2, 3, x ∈ X}

R2 ={(

xi , xi+2 (mod 3)

) ∣∣ i = 1, 2, 3, x ∈ X}

R3 = E

R4 ={

(xi , yj)∣∣ i = 1, 2, 3, {xi+1 (mod 3), yj} ∈ E

}R5 =

{(xi , yj)

∣∣ i = 1, 2, 3, {xi+2 (mod 3), yj} ∈ E}




Remark

Notice that the relations R1,R2,R4,R5 are anti-symmetric andthat R1 = Rt

2 and R4 = Rt5.


A6 (Γ) :=(V , {Ri}5i=0

)is a commutative association scheme.

The following proposition implies that this is a sufficient condition:


Every rank 6 association scheme with parameters as inConstruction 2 arises from a regular 3-graph.




Combining the above results, we formulate the followingcharacterization of regular 3-graphs:

Corollary

Let Γ be an (n, 3, c2)-cover. The following are equivalent:

1 Γ defines a regular 3-graph.

2 Γ has cyclic matchings.

3 Γ is a cyclic cover.

4 A6(Γ) is an association scheme.




Finding feasible parameter sets of regular 3-graphs

Three sources:

(i) the association schemes A4(Γ) and A6(Γ) (theircharacter-multiplicity tables),

(ii) eigenvalues and multiplicities of the weight (viewed as amatrix),

(iii) Godsil-Hensel results about cyclic ADRG’s.




Proposition

Necessary conditions for the set (n, a, b) of parameters of a regular3-graph are:

(i) 3|n,

(ii) n = a + 2b + 2,

(iii) The roots α and β of the equationx2 − (a− b)x − (n − 1) = 0 are integers,

(iv) n − 1 + α2 divides 2n(n − 1),

(v) α− β divides nα.




A corrected version of Higman’s feasibility conditions yields a list137 parameter sets with n ≤ 1000. With our additional condition,this list boils down to a list of 64 parameter sets. We haveconstructions for 6 parameter sets, most of them are new examplesof regular 3-graphs.




Layout



3 Regular 3-graphs

4 New constructions




The Klin-Pech construction

In their recent paper, Klin and Pech provide a construction ofcyclic (n2, 3, n

2

3 )-covers from a generalized Hadamard matrices oforder n. We used classifications of generalized Hadamard matriceswith suitable parameters to construct all non-isomorphic cycliccovers, which provide different regular 3-graphs. A summary of ournew constructions of regular 3-graphs:

1 new example with parameters (36, 10, 12),

1 new example with parameters (45, 19, 12) (mistake in thesistext),



28 new examples with parameters (324, 106, 108).




About the mistake with 135 vertices

To check if an ADRG defines a regular 3-graph, I wrote a GAPfunction that takes as input an ADRG, and constructs the relationsR0, . . .R5 which appear in Construction 2. This function thenchecks whether the obtained set of relations forms an associationscheme.There was critical mistake in my implementation of this function:a concrete labelling was assumed, that is, it was assumed that theantipodal fibers are labelled sequentially.The result of this mistake was that the answer the functionreturned for the (45, 3, 12)-cover was false. This led to areformulation of the equivalence of “cyclic cover” with “cover withcyclic matchings”.




The example with parameters (36, 10, 12)

There is a only one gH(U3, 6) up to monomial equivalence. It is:

H =

1 1 1 1 1 11 1 ζ ζ2 ζ2 ζ1 ζ 1 ζ ζ2 ζ2

1 ζ2 ζ 1 ζ ζ2

1 ζ2 ζ2 ζ 1 ζ1 ζ ζ2 ζ2 ζ 1

.




Using the Klin-Pech method, I constructed the correspondingcyclic (36, 3, 12)-cover Γ. It is not distance transitive, itsautomorphism group is transitive of rank 8 on the vertices of thecover, and is isomorphic to (Z3.S6).Z2. Then I constructed A6(Γ),which turned out to be an association scheme, producing a newexample of a regular 3-graph. It is a regular 3-graph withparameters (36, 10, 12).




The equivalence of gH matrices and STD’s

The fact that generalized Hadamard matrices are equivalent toclass-regular symmetric transversal designs led us to an attempt toconstruct the ADRG in simple terms of the STD. We present as aprototypical example the STD2(3) and the corresponding(36, 3, 12)-cover discovered by Klin and Pech. We present our newmodel of this design starting from the alternating group A5.




A model of the STD2(3) via A5

Consider the following complementary graphs on 5 vertices:

The Pentagon

•1

•2

•3

•4

•5 HHHHHHHHHH

��))))))))))

vvvvvvvvvv

C

The Pentagram

•1

•2

•3

•4

•5

)))))))))))))))))

HHHHHHHHHHHHHHHHHvvvvvvvvvvvvvvvvv

��

S




Now consider the action of the alternating group A5 of order 60 onthese graphs, specifically, consider the orbits CA5 and SA5 of Cand S . Notice that in both cases the stabilizer SA5(C ) is theautomorphism group Aut(C ) of C (and thus also of S), which isthe dihedral group D5 of order 10. This means that the orbits areof size 6. We depict them:




•1

•2

•3

•4

•5 HHHH

��))))

vvvv

c1

•1

•2

•3

•4

•5 HHHH

))))))

vvvvvvHHHHHH))))

c2

•1

•2

•3

•4

•5

��vvvv

��vvvvvvHHHHHH

c3

•1

•2

•3

•4

•5 HHHH

��HHHHHH

c4

•1

•2

•3

•4

•5))))))

vvvv

vvvvvv

c5

•1

•2

•3

•4

•5))))))

��

))))

c6

•1

•2

•3

•4

•5))))))

��

vvvvvvHHHHHH

s1

•1

•2

•3

•4

•5

��vvvv

��

s2

•1

•2

•3

•4

•5 HHHH

))))))))))

s3

•1

•2

•3

•4

•5))))))

vvvv��vvvvvv))))

s4

•1

•2

•3

•4

•5 HHHH

��

��HHHHHH))))

s5

•1

•2

•3

•4

•5 HHHHvvvv

vvvvvv

HHHHHH

s6




Also, consider the orbit of A5 on the pair {C , S}. The stabilizerSA5({C , S}) is still D5 since a permutation sending C to S (forexample, the permutation (2, 4, 5, 3)) is odd. Clearly, this orbit is

{C , S}A5 = {{ci , si}|1 ≤ i ≤ 6} .

The set CA5 ∪ SA5 ∪ {C ,S}A5 is our set of 18 points of theSTD2(3). Now we define the blocks. There are two types ofblocks:

3 blocks of type 1,

15 blocks of type 2.




Type 1 blocksThese are the 3 blocks consisting of the sets CA5 , SA5 and{C ,S}A5 .Type 2 blocksConsider the subgraphs of K5 of the following type:

•

•

••

• ��

))))))

A graph of type γ




There are 15 = 5 · 3 such subgraphs, to each such (labeled)subgraph we associate the following 6 points:

••

••

• HHHH

��))))

vvvv••

••

•)))))))

��

��

))))

••

••

•)))))))

vvvv

��vvvvvvv))))

••

••

• HHHH

��

��HHHHHHH))))

••

••

• HHHH

)))))))

vvvvvvvHHHHHHH

))))

••

••

•��vvvv

��

••

••

•��vvvv

��

vvvvvvvHHHHHHH

••

••

• HHHH

)))))))

))))

A block of type 2




Explicitly, the block associated with such a graph consists of

the 2 points in CA5 containing this graph as a subgraph,

the 2 points in SA5 containing this graph as a subgraph,

the 2 points in {C , S}A5 for which neither element of the paircontains this graph as a subgraph.

The incidence in this structure is natural inclusion.




The parallel classes

The 6 parallel classes of points are:

{{ci , si , {ci , si}}|1 ≤ i ≤ 6}.The 6 parallel classes of blocks are of two types, 1 parallelclass is the set of type 1 blocks. Each of the other 5 parallelclasses is determined by the 3 different graphs of type γcorresponding to a fixed isolated vertex. Explicitly, for thevertex x , the corresponding parallel class of blocks is:

•x

•

••

•

��))))))

•x

•

••

•

•x

•

••

•

vvvvvvvvvvHHHHHHHHHH




A construction of ADRG’s from TD’s

An ambitious task was to understand the construction of Klin andPech without the use of generalized Hadamard matrices. In otherwords, we tried to construct the ADRG in terms of the TD. Forthis purpose we introduced the language of the flag algebra (L- andN- walks, distances etc.) of designs, as it was developed by Pech.We have made some progress in this direction in the last weeks.




The following figure (due to Pech) describes the L-distance andN-distance distribution with respect to the orbit partition of thestabilizer of a distinguished flag.The ADRG is obtained by merging the following classes:

5 flags with L-distance 1,

5 flags with N-distance 1,

5 flags with L-distance=N-distance=2,

20 out of the 30 flags with L-distance=N-distance=3.

Thus our goal was to characterize the last 20 flags in terms offlags.There are 5 flags with L-distance=N-distance=2 from each of the2 other flags in this fiber, these form the class of 10 flags.




1

5

5

5

20

20

10

20

10

2

10

N???????????

L

��

N

???????????

L

��N

N

��

L ??????????

L

LOOOOOOO

L

444444444444444N

Nooooooo

N

L

L ooooooo

N

444444444444444 L

N OOOOOOOL

��

N

??????????L N

N

L

L,N

N,L

L,N

L

N

L-distance and N-distance distribution




After this characterization, we can describe A6(Γ) in terms of theflags of the STD.

35

35

35

1

1

1

qqqqqqMMMMMM

111

A6(Γ)


Thesis exam Regular t-graphs, antipodal distance regular ...dannykal/research/thesis exam...

Documents

Transcript of Thesis exam Regular t-graphs, antipodal distance regular ...dannykal/research/thesis exam...