Graph theory: fundamentals and applications to crystallographic and

crystallochemical problems.The vector method

Jean-Guillaume Eon - UFRJ

ECM-23, 4-6 August 2006, Leuven

Summary

• Motivation• Fundamentals of graph theory• Crystallographic nets and their quotient graphs• Space group and isomorphism class• Topological and geometric properties

Objectives

• Topology and Geometry: Description and Prevision of Crystal Structures

• Topological properties: Crystallographic Nets and Quotient Graphs

• Geometric properties: Crystal Structures as Embeddings of Crystallographic Nets

Crystal structures as topological objects

Crystal 1-Complex Net

Chemicalobject

Geometric object

Topological object

Atom Point Vertex

Bond Line Edge

Example: ReO3

1-Complex Crystallographic net= Embedding of the

Graph theory (Harary 1972)

A graph G(V, E, m) is defined by:– a set V of vertices,– a set E of edges,– an incidence function m from E to V2

u

v

w

m (e1) = (u, v)

m (e2) = (v, w)

m (e3) = (w, u)

V = {u, v, w} E = {e1, e2, e3}

e1e2

e3

e1 = uve1

-1 = vu

Simple graphs and multigraphs

loopMultiple edges

Simple graph: graph without loops or multiple edges

Multigraph: graph with multiple edges

Some elementary definitions

• Order of G: |G| = number of vertices of G• Size of G: ||G|| = number of edges of G• Adjacency: binary relation between edges or vertices• Incidence: binary relation between vertex and edge• Degree of a vertex u: d(u) = number of incident edges

to u (loops are counted twice)• Regular graph of degree r: d(u) = r, for all u in V

Adjacentvertices

Adjacentedges

Incidence relationship

Walks, paths and cycles• Walk: alternate sequence of incident vertices

and edges

• Closed walk: the last vertex is equal to the first = the last edge is adjacent to the first one

• Trail: a walk which traverses only once each edge

• Cycle: a closed path

• Forest: a graph without cycle

• Path: a walk which traverses only once each vertex

a.b.j.i trail

a.b.j.d path

b.j.d.e.f.g.h.i closed trail b.j.i cycle

Edges only are enough to define the walk!

b.j.i.b.c walk

b.j.i.b.j.i closed walk

Nomenclature

• Pn path of n edges• Cn cycle of n edges• Bn bouquet of n loops• Kn complete graph of n vertices• Kn

(m) complete multigraph of n vertices with all edges of multiplicity m

• Kn1, n2, ... nr complete r-partite graph with r sets of ni vertices (i=1,..r)

C5

P3

B4

K3(2)K4

K2, 2, 2

Connectivity

• Connected graph: any two vertices can be linked by a walk

• Point connectivity: κ(G), minimum number of vertices that must be withdrawn to get a disconnected graph

• Line connectivity: λ(G), minimum number of edges that must be withdrawn to get a disconnected graph

Determine κ(G) and λ(G)

G

λ(G) = 3

κ(G) = 2

Subgraph of a graphComponent = maximum connected subgraphTree = component of a forest

T1 T2

G = T1 T2

Spanning graph = subgraph with the same vertex set as the main graph

G

T: spanning tree of G

T

Hamiltonian graph: graph with a spanning cycle

Show that K2, 2, 2 is a

hamiltonian graph

Eulerian graph: graph with a closed trail “covering” all edges

Show that K2, 2, 2 is an Eulerian graph

Distances in a graph

•Length of a walk: ||W|| = number of edges of W

•Distance between two vertices A and B: d(A, B) length of a shortest path (= a geodesic)

A

B A---B|=3|A---B|=4d(A, B)=3

Morphisms of graphs

• A morphism between two graphs G(V, E, m) and G’(V’, E’ m’) is a pair of maps fV and fE between the vertex and edge sets that respect the adjacency relationships:

for m (e) = (u, v) : m’ {f E (e)} = ( f V (u), f V (v))

or: f(uv) = f(u)f(v)

• Isomorphism of graphs: 1-1 and onto morphism

e3

e1

e2

v

w

C3

f

u

b

a

e4

e5

f(u) = f(v) = a f(w) = b

f(e1) = e4 f(e2) = f(e3) = e5

Example: an onto morphism in non-oriented graphs

f(vw) = f(v)f(w)

Aut(G): group of automorphisms

• Automorphism: isomorphism of a graph on itself.

• Aut(G): group of automorphisms by the usual law of composition, noted as a permutation of the (oriented) edges.

v

u

w

e1e2

e3

C3Generators for Aut(C3) order 6

(e1, e2, e3) order 3

(e1, e3-1) (e2, e2

-1) order 2{

Quotient graphs

• Let G(V, E, m) be a graph and R < Aut(G)• V/R = orbits [u]R of V by R• E/R = orbits [uv]R of E by R• G/R = G(V/R, E/R, m*) quotient graph, with

m* ([uv]R) = ([u]R, [v]R)

• The quotient map: qR(x) = [x]R (for x in V or E), defines a graph homomorphism.

e3

e1

e2

u v

w

R: generated by (e1, e2, e3)( noted R = <(e1, e2, e3)> )

C3

[u]R

[e1]R

B1 = C3/R

qR

u v

wx

e1

e2

e3

e4

R = <(u, v)(x, w)>

Find the quotient graph C4/R

C4[u]R

[x]R[e2]R

[e3]R

[e1]R

qR

Cycle and cocycle spaces on Z

• 0-chains: L0={∑λiui for ui V and λi in Z}• 1-chains: L1={∑λiei for ei E and λi in Z}• Boundary operator: ∂e = v – u for e = uv• Coboundary operator: δu = ∑ei for all ei = uvi

• Cycle space: C = Ker(∂) (cycle-vector e : ∂e = 0)• Cocycle space: C* = Im(δ) (cocycle-vector c = δu)• dim(cycle space): = ||G|| - |G| +1 (cyclomatic

number)

Cycle and cocycle vectors

A

B

C

D

1 2

3

4

56

Cycle-vector: e1+e2-e3

Cocycle-vector: δA = -e1+e2+e4

∂e1 = A - D

Cycle and cut spaces on F2 = {0, 1}(non oriented graphs)

• The cycle space is generated by the cycles of G• Cut vector of G = edge set separating G in

disconnected subgraphs• The cut space is generated by δu (u in V)

u

v

w

Find δ(u + v + w)

u

v

wδu

u

v

w

u

v

w δu + δv

u

v

w δu + δv

u

v

w

u

v

w δu + δv + δw

u

v

w

Complementarity: L1 = C C*

• Scalar product in L1: ei.ej = ij

• dim(C*) = |G|-1

• dim(C) + dim(C*) = ||G||

• C C*

uC6

C6 . u = 0

Example: K2(3)

A B

e1

e2

e3

L1: e1, e2, e3 Natural basis (E)

C: e2 – e1, e3 – e2

C*: e1 + e2 + e3} Cycle-cocycle basis (CC)

CC = M.E M =-1 1 0 0 -1 1 1 1 1

Write the cycle-cocycle matrix for the graph below

A

B C

e1

e2 e3

e4

e5

M =

1 -1 0 0 0 0 1 -1 0 1 0 0 1 -1 0 1 1 1 1 0-1 -1 0 0 1

Periodic graphs

• (G, T) is a periodic graph if:

• G is simple and connected,

• T < Aut(G), is free abelian of rank n,

• T acts freely on G (no fixed point or edge),

• The number of (vertex and edge) orbits in G by T is finite.

Example: the square net

V = Z2

E = {pq | q-p = ± a, ± b}

a = (1,0), b = (0,1)

T = {tr: tr (p) = p+r, r in Z2}

Invariants in perodic graphs:1. rings

• Topological invariants are conserved by graph automorphisms

• A ring is a cycle that is not the sum of two smaller cycles

• A strong ring is a cycle that is not the sum of (an arbitrary number of) smaller cycles

Example 1: the square net

A cycle (with shortcuts)

Sum of three strong rings

A strong ring

Example 2: the cubic net

Strong rings:

Cz = ABCDA Cx = BEFCB Cy = CFGDC

A

B C

D

E F

G

A cycle that is not a ring: CBEFGDC

A

B C

D

E F

G

CF = shortcut

= Cx + Cy

A ring that is not a strong ring: ABEFGDA

No shortcut!

A

B

D

E F

G

C

= Cx + Cy + Cz

Invariants in periodic graphs: 2. Infinite geodesic paths

• Given a graph G, a geodesic L is an infinite, connected subgraph of degree 2, for which, given two arbitrary vertices A and B of L, the path AB in L is a geodesic path between A and B in G.

• L is a strong geodesic if the path AB in L is the unique geodesic between A and B in G.

Example 1: the square net

Strong geodesics

Geodesics

Yi

Xj

Infinite path (shortcut)

Are the following infinite paths geodesicsor strong geodesics?

Strong geodesic

Example 2: the β-W net

Strong geodesic

Geodesic

Example 3: the 34.6 net

Not every periodic graph has strong geodesics!

Geodesic fiber

• 1-periodic subgraph F of a periodic graph G, such that:

• for any pair of vertices x and y, F contains all geodesic lines xy of G, and

• F is minimal, in the sense that no subgraph satisfies the above conditions.

Example: the β-W net

The 34.6 net

Crystallographic nets

• Crystallographic net: simple 3-connected graph whose automorphism group is isomorphic to a crystallographic space group.

• n-Dimensional crystallographic space group: a group Γ with a free abelian group T of rank n which is normal in Γ, has finite factor group Γ/T and whose centralizer coincides with T.

Local automorphisms

• Automorphism f with an upper bound b for the distance between a vertex and its image:

d{u, f (u)} < b for all u• The set L(N) of local automorphisms of an infinite

graph N forms a normal subgroup of Aut(N)• A net is a crystallographic net iff the group of

local automorphisms L(N) is free abelian and admits a finite number of (vertex and edge) orbits

Automorphisms and geodesics

• Graph automorphisms map (strong) geodesics on (strong) geodesics.

• A strong geodesic is said to be along f if it is invariant by the local automorphism f.

• Two strong geodesics are parallel if they are invariant by the same local automorphism.

• Local automorphisms map strong geodesics on parallel ones.

(0,0)

Example: the square net

(i, j)

Conversely: Let f be any local automorphism with f(0,0) = (i,j); then, we will see that ti, j

-1. f = 1

T={ti,j: ti,j (m, n) = (m+i, n+j)} : T L(N)

Y0 Yi

X0

Xj

Y0 invariant by t0,1 as Yi

X0 invariant by t1,0 as Xj

fixes X0 and Y0

fixes the whole net

L(N) = T

Example: the square net

Simple,

Every automorphism of the infinite graph can beassociated to an isometry of the plane

Crystallographic net

4-connected

L(N) = T free abelian

One vertex orbit

Two edge orbits

Quotient graph of a periodic graph: the square net

Q = G/T, T translation subgroup T of the automorphism group

Example: the β-W net

Show that β-W is a crystallographic net andFind its quotient graph

at

bt ct

t in Z2, i=(1,0), j=(0,1)

V = {at , bt , ct}

E = {atbt , atct , btct , atbt+j , atct+j , ctbt+i}

β-W net: quotient graph

a

b c

ReO3: vertex and edge lattices

O 1 O 1O 2

O 2

O 3

O 3

R e e1e2

Quotient graph: ReO3

O 1

e 2 e 1

O 2O 3

e 3

e 4 e 5

e 6R e

K1,3(2) K1,3

(2)

Structure Quotient graph

NaCl K2(6)

CsCl K2(8)

ZnS (sphalerite) K2(4)

SiO2 (quartz) K3(2)

CaF2 (fluorite) K1,2(4)

Simple structure types

Symmetry

: Space group of the crystal structure

• T: Normal subgroup of translations/T: Factor group

• Aut(G): Group of automorphism of the quotient graph G

/T is isomorphic to a subgroup of Aut(G)

ReO3: /T Aut(G) m3m

O 1 O 1O 2

O 2

O 3

O 3

R e e1e2

• C4 Rotation: (e1, e3, e2, e4)• C3 Rotation: (e1, e3, e5) (e2, e4, e6)

• Mirror: (e1, e2)

O 1

e 2 e 1

O 2O 3

e 3

e 4 e 5

e 6R e

Drawbacks of the quotient graph

• Non-isomorphic nets can have isomorphic quotient graphs

• Different 1-complexes with isomorphic nets can have non-isomorphic quotient graphs

• Different 1-complexes can have both isomorphic nets and isomorphic quotient graph!

A

BC

D

A AB B

C

C

D

D

Different nets with isomorphic quotient graphs

A

BC

D

AAB

B

C

C

D

D

A

D

BC

A (0 1 )

B (1 0 )

0 1

1 0

b

a

b

a

1 00 1

Vector method: the labeled quotient graph as a voltage graph

β-W net: Find the labeled quotient graph

at

bt ct

t in Z2, i=(1,0), j=(0,1)V = {at , bt , ct}E = {atbt , atct , btct , atbt+j , atct+j , ctbt+i}

A

B C10

01 01

The minimal net

• Translation subgroup isomorphic to the cycle space of the quotient graph G (dimension equal to the cyclomatic number)

• Factor group: isomorphic to Aut(G)• Unique embedding of maximum symmetry:

the archetype (barycentric embedding for stable nets)

Example 1: graphite layer - K2(3)

1

2

3

A B

1-chain space: 3-dimensional

Two independent cycles: a = e1-e2, b = e2-e3

One independent cocycle vector: e1+ e2+e3

e1

e2

e3

ab

1

2

3

A B

Generators of Aut[K2(3)]:

• (e1, e2, e3)• (ei , -ei)(A, B)• (e1, -e3)(e2, -e2)(A, B)

Linear operations in the cycle space (point symmetry):

(a, b) = (e1 – e2, e2 – e3) → (e2 – e3, e3 – e1) = (b, - a - b)

In matrix form: 0 -1

1 -1γ{(e1, e2, e3)} =

0 1

1 0γ{(e1, -e3)( e2, -e2)} =

-1 0

0 -1γ{(ei, -ei)} =

p6mm

Embedding in E3

Inversion of the cycle-cocycle Matrix

a = e1 – e2

b = e2 – e3

c = e1 + e2 + e3

a 1 -1 0 e1

b = 0 1 -1 x e2

c 1 1 1 e3

e1 2 1 1 ae2 = 1/3 -1 1 1 x be3 -1 -2 1 c

Barycentric embedding: c = 0

e1 = 1/3 (2a + b)e2 = 1/3 (-a + b)e3 = 1/3 (-a – 2b)

Atomic positions

p6mmGenerator 0 -1 -1 0 0 1 1 -1 0 -1 1 0Translation (0 0) (0 0) (0 0)

A: x + TB: x + e1 + T

After inversion: A → -x + T = B = x + 2a/3 + b/3 + T-2x = 2a/3 + b/3 + TWith t = b , x = -a/3 -2b/3 = 2a/3 + b/3

A: 2a/3 + b/3 + TB: a/3 + 2b/3 + T

Example 2: hyper-quartz

1

2

34

5

6

Archetype: 4-dimensionalTetrahedral coordenationCubic-orthogonal family:(e1-e4), (e2-e5), (e3-e6), ∑ei

Lengths: √2, √2, √2, √6Space group: 25/08/03/003

Nets as quotients of the minimal net

100

010 001

<111> = translation subgroup generated by 111 N/<111>: T = <{100, 010}>

N (T = Z3) N/<111>-1 -1 0

ab

(Periodic) voltage net

ab

100010

= 1 1 1

Embeddings as projections of the archetype

A

B

C

D

1 2

3

4

56

3-d archetype,2-d structures: 1-d kernel

3-cycle: 3.92, 93 4-cycle: 4.82

A

BC

D0 1

1 0

A

BC

D0 0 1

0 1 0

1 0 0

A

D

BC

1 00 1

A

D

BC

0 1 00 0 1

1 0 0

S r[S i ]

( 4 3 2 )2

I1

(4 .8 )4

2

p m m(3 .9 ,9 )

3 1

2 3

p m

Check the space group of 4.82

A

B

1 2

3

4

56

DC

Kernel: c = e4 + e6 + e3 – e2

Generators:(e4,e6,e3,-e2)(e1,-e5,-e1,e5)(A,B,C,D)(e4,-e6)(e2,e3)(e1,-e1)(A,C)

Cycle basis:a = e1 + e4 + e5 – e3

b = e1 + e2 - e5 + e6

Notice: a ┴ b ┴ c

Point symmetry

(a, b) → (-e5 + e6 + e1 + e2, -e5 – e4 – e1 + e3) = (b, -a)

(a, b) → (-e1 – e6 + e5 – e2, -e1 + e3 - e5 – e4) = (-b, -a)

p4mmGenerator 0 -1 0 -1 1 0 -1 0 Translation (0 0) (0 0)

Walk: A → B → C → D → AFinal translation : c = 0

e4 e6 e3 -e2

Find the space group of quartz

1

2

34

5

6

a = 1000 = e1 - e4

b = 0100 = e2 - e5

c = 0010 = e3 - e6

d = 0001 = e4 + e5 + e6

Hyper quartz = N[K3(2)]

Quartz = N[K3(2)]/<1110>

A

B C

Point symmetry

Generators: (1,4)(2,5)(3,6) order 2 (1,2,3)(4,5,6)(A,B,C) order 3 (1,6)(2,5)(3,4)(B,C) order 2

(e1 + e2 + e3 - e4 - e5 - e6 = 0 )

Basis: (a, b, d)

order 12

: (a, b, d) (-a, -b, d) C2 rotation d : (a, b, d) (b, -a-b, d) C3 rotation d: (a, b, d) (-a-b, b, -d) C2 rotation d

C6 rotation

Translation part

Associated to the C3 rotation:

Walk: A → B → C → A

Final translation : e1+e2+e3 = d 31 or 62

e1 e2 e3

For and , t=0 (fixed points) P6222

Isomorphism class of a net

ab

Find the labeled quotient graph for the double cell

a bb

Isomorphism class of a net

4, a

bb

1 3 2

Automorphisms of the quotient graph G, which leave the quotient group of the cycle space by its kernel invariant, can be used to extend the translation group if they act freely:

t(o.q-1o [W]) = o.q-1

o [.(W)] ,where δ is a path in G from O to (O)

Kernel: e1-e2

: (e1, e2)(e3, e4)

Isomorphism class of a nett(o. q-1

o [W]) = o.q-1o [.(W)]

t: new translationo: arbitrary origin in N: boundary operatorq: N G = N/TW: walk in G starting at O=q(o)q-1

o [W]: walk in N starting at o and projecting on Wo. q-1

o [W]: defines a vertex of Nδ: a fixed, arbitrary path in G from O to (O): (extension) automorphism of the quotient graph G

W = OO (path nul): t(o) = o.q-1o

[]

Isomorphism class of a net

4, a

bb

1 3 2

Kernel: e1-e2

Extension by = (e1, e2)(e3, e4)

b

A B

{A, B}

{e1, e2} {e3, e4}

With o in q-1(A) and = e3 : t (o) = o.q-1o [e3]

t 2 (o) = t (o.q-1o [e3]) = o.q-1

o [e3.(e3)] = o.q-1o [e3.e4] = a(o)

t2 = at ?

ab

ab

t:

t(o. q-1o [W]) = o.q-1

o [.(W)]

Show that cristobalite has the diamond net

100

101

001

010010

e1

e8e7

e6e5e4e3

e2

A

C

B

D

Kernel: e2 – e1 = e7 – e8

e3 – e4 = e6 – e5

(1,8)(2,7)(3,6)(4,5)(A,D)(B,C)

{A,D} {B,C}

{1,8}

{2,7}, 100

{4,5}, k

e8 – e4 + e1 – e5 = 001 → e1 – e5 + e8 – e4 = 001

2k = e4 – e8 + e5 – e1 = 0 0 -1

{3,6}-{4,5} = 010 l = 010 + k

{3,6}, l

Non-crystallographic nets

Local automorphisms = T x {I, σ} : abelian but not free!

T

σ

T = <t: t (xm) = xm+1 , x = a, b>

a

b

t te

A Bσ: (e, -e)

Non-crystallographic nets

Local automorphisms = T’ x {I, σ}

T’

σ

T’ = <t’: t’ (am) = bm+1 , t (bm) = am+1 >

a

b

t’

-t’

tfc and derived nets

A

B C001

100 010

A

B C

10 01

tfc/<(001)>

A

B C

1 1

tfc / < (001), (110) >

Local automorphisms = not abelianBut : (b0, c0) ↔ (b, c)

b0

c0

T

The 1-periodic graph is a geodesic fiber

Projection of a geodesic fiber on the quotient graph: the β-W net

at

bt ct

A

B C10

01 01

F

G G/T

F/<01> is asubgraph of G/T

01 01

The 34.6 net (G/T = K6)

10

10

10

11

(Counterclockwiseoriented cycles)

21

G 21

G 10

10-1 0

-1 010

Determine the quotient graph G/T of the 32.4.3.4 net,

list all the cycles of G/T; find the strong geodesic lines

of 32.4.3.4 and its geodesic fibers.

a

d

b

c

A

D

B

C

10

10

10 01

0101

2-cyclesAB, CD: 10AD, BC: 01

3-cyclesABC: 00, 01, 11, 10ABD: 00, 01, -1 0, -1 1ACD: 00, -1 0, 0 -1, - 1 -1BCD: 00, 0 -1, 1 -1, 10

4-cyclesABCD: 00, 01, 10, 11, 1 -1ABDC: 01, 11, -1 1ADBC: 1 -1, 10, 11

Strong geodesic lines: 10, 01

disjoint cycles

tangent cycles

Geodesic fibers: 11, -1 1

a

d

b

c

A

D

B

C

10

10

G 10

A

D

B

C

10 01

G 11

01

a

d

b

c01

A

D

BC

11

G 11

a

d

b

c

Geodesic fibers and local automorphisms

• Local automorphisms send geodesic fibers to parallel geodesic fibers.

• The absence, in 2-connected labeled quotient graphs, of automorphism that leave unchanged the vector labels of any loop or cycle ensures the derived graph is a crystallographic net.

Cycle figure

10

0 -1

10

01 11

-1 0

0 -1-1 -1

Def: vector star in the direction of geodesic fibers

Graphite layer

Topological density

ρ = lim ∑k=1,r Ck / rn

Ck: kth coordination numbern : dimension of the net

r→∞

ρ = Z.{Σσ f(σ)}/ n! Z : order of the quotient graphf(σ) : frequency of the face σ of the cycles figure= inverse product of the lengths of the verticesof the face of the cycles figure

Topological density: graphite layer

10

01 11

-1 0

0 -1-1 -1

σ1

f(σ1) = 1/(2.2)

ρ = Z.{Σσ f(σ)}/ n!

ρ = 2.6.(2.2)-1/2! = 3/2

Topological density: net 32.4.3.4

10-1 0

01

0 -1

11-1 1

-1 -1 1 -1

3-cycle

2-cycle

ρ = 4.8.(2.3)-1/2! = 8/3

Find the topological density the β-W net

B C

01 01

A

1010

01

0 -1

11-1 1

-1 -1 1 -1

3-cycle

2-cycle

ρ = 3.8.(2.3)-1/2! = 2

Plane nets

Embedding of the quotient graph in the torus

A

B C

10 01

a

a

bb

Face boundaries project on oriented trails ofthe quotient graph

Oriented trails are based on oriented edges

e1, 10 e2, 01

A

B C

a b10

c

a0-1 b1-1a1-1

c10

e1, 10 e2, 01

A

B C

K4 revisited

3.92, 93 4.82

Two classes of trails: 3-cycles and 4-cycles

A plane net from K3, 3

Cyclomatic number:υ = 9 – 6 + 1 = 4

Kernel : two tangent 4-cycles

a

a

b b

4.102, 42.10

Topological invariants of nets in projection on the quotient graph

• Strong geodesic• Geodesic fiber

• Strong ring

• Shorter isolated cycle• Connected subgraph of

shorter cycles (of equal length) with equal or opposite net voltages

• Set of edge-disjoint cycles with nul net voltage

Non-ambiently isotopic embeddings of the same net

ThSi2 SrSi2

Self-interpenetrated ThSi2

Self-interpenetrated SrSi2

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