Jean-Guillaume Eon - UFRJ

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ECM-23, 4-6 August 2006, Leuven. Graph theory: fundamentals and applications to crystallographic and crystallochemical problems . The vector method. Jean-Guillaume Eon - UFRJ. Summary. Motivation Fundamentals of graph theory Crystallographic nets and their quotient graphs

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ECM-23, 4-6 August 2006, Leuven

### Graph theory: fundamentals and applications to crystallographic and crystallochemical problems.The vector method

Jean-Guillaume Eon - UFRJ

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
Example: ReO3

1-Complex

= Embedding of the

Crystallographic net

v

e2

e1

u

e3

w

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

V = {u, v, w}

E = {e1, e2, e3}

m (e1) = (u, v)

m (e2) = (v, w)

m (e3) = (w, u)

e1 = uv

e1-1 = vu

Simple graphs and multigraphs

loop

Multiple edges

Simple graph: graph without loops or multiple edges

Multigraph: graph with multiple edges

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

edges

vertices

Incidence relationship

Some elementary definitions
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
• Path: a walk which traverses only once each vertex
• Cycle: a closed path
• Forest: a graph without cycle

b.j.i.b.c walk

a.b.j.i trail

a.b.j.d path

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

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

b.j.i cycle

Edges only are enough to define the 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)

P3

C5

B4

K4

K3(2)

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

G

κ(G) = 2

λ(G) = 3

Determine κ(G) and λ(G)

Subgraph of a graph

Component = maximum connected subgraph

Tree = component of a forest

T2

T1

G = T1 T2

G

T

T: spanning tree of G

Hamiltonian graph: graph with a spanning cycle

Show that K2, 2, 2 is a

hamiltonian graph

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|=3

B

|A---B|=4

A

d(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’ {fE (e)} = ( f V (u), fV (v))

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

• Isomorphism of graphs: 1-1 and onto morphism

Example: an onto morphism in non-oriented graphs

w

b

e5

f

e3

e2

a

C3

u

v

e4

e1

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

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

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

v

C3

{

e2

e1

(e1, e2, e3) order 3

u

(e1, e3-1) (e2, e2-1) order 2

e3

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.

Generators for Aut(C3) order 6

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.

w

[e1]R

qR

e3

e2

C3

v

u

[u]R

e1

B1 = C3/R

R: generated by (e1, e2, e3)

( noted R = <(e1, e2, e3)> )

[e3]R

e3

x

w

qR

[x]R

e4

e2

[e2]R

C4

[u]R

u

v

e1

[e1]R

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

Find the quotient graph C4/R

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

∂e1 = A - D

1

4

2

Cycle-vector: e1+e2-e3

Cocycle-vector: δA = -e1+e2+e4

C

6

5

D

3

B

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)

w

u

v

Find δ(u + v + w)

w

u

v

δu

w

u

v

w

u

v

δu + δv

w

u

v

δu + δv

w

u

v

w

u

v

δu + δv + δw

w

u

v

u

C6

C6 . u = 0

Complementarity: L1 = C  C*
• Scalar product in L1: ei.ej = ij
• dim(C*) = |G|-1
• dim(C) + dim(C*) = ||G||
• C  C*
Example: K2(3)

e1

e2

B

A

e3

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

}

C: e2 – e1, e3 – e2

Cycle-cocycle basis (CC)

C*: e1 + e2 + e3

-1 1 0

0 -1 1

1 1 1

M =

CC = M.E

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

M =

Write the cycle-cocycle matrix for the graph below

A

e1

e4

e2

e3

B

C

e5

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

A strong ring

Example 1: the square net

A cycle (with shortcuts)

Sum of three strong rings

Example 2: the cubic net

A

D

Strong rings:

Cz =ABCDA

Cx = BEFCB

Cy = CFGDC

B

C

G

E

F

= Cx + Cy

A cycle that is not a ring:

CBEFGDC

A

D

B

C

CF = shortcut

G

E

F

= Cx + Cy + Cz

A ring that is not a strong ring:

ABEFGDA

A

D

B

C

No shortcut!

G

E

F

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

Yi

Infinite path (shortcut)

Geodesics

Xj

Strong geodesics

Example 2: the β-W net

Strong geodesic

Are the following infinite paths geodesics

or strong geodesics?

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.
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.
Example: the square net

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

Yi

Y0

Y0 invariant by t0,1 as Yi

X0 invariant by t1,0as Xj

Xj

(i, j)

•  fixes X0 and Y0
•  fixes the whole net
• L(N) = T

X0

(0,0)

Conversely: Let f be anylocal automorphism with f(0,0) = (i,j); then, we will see that ti, j-1. f = 1

Example: the square net

Simple,

4-connected

L(N) = T free abelian

One vertex orbit

Two edge orbits

Every automorphism of the infinite graph can be

associated to an isometry of the plane

Crystallographic net

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

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

V = {at , bt , ct}

E = {atbt , atct , btct , atbt+j ,

atct+j , ctbt+i}

at

bt

ct

Show that β-W is a crystallographic net and

Find its quotient graph

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

01

01

B

C

10

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}

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)

b

a

Example 1: graphite layer - K2(3)

1

A

B

e1

2

e2

3

e3

1-chain space: 3-dimensional

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

One independent cocycle vector: e1+ e2+e3

0 -1

1 -1

γ{(e1, e2, e3)} =

In matrix form:

0 1

1 0

-1 0

0 -1

γ{(e1, -e3)( e2, -e2)} =

γ{(ei, -ei)} =

• Generators of Aut[K2(3)]:
• (e1, e2, e3)
• (ei , -ei)(A, B)
• (e1, -e3)(e2, -e2)(A, B)

1

A

B

2

3

Linear operations in the cycle space (point symmetry):

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

p6mm

a 1 -1 0 e1

b = 0 1 -1 x e2

c 1 1 1 e3

e1 2 1 1 a

e2 = 1/3 -1 1 1 x b

e3 -1 -2 1 c

e1 = 1/3(2a + b)

e2 = 1/3 (-a + b)

e3 = 1/3 (-a – 2b)

Barycentric embedding: c = 0

Embedding in E3

Inversion of the cycle-cocycle Matrix

a = e1 – e2

b = e2 – e3

c = e1 + e2 + e3

Atomic positions

p6mm

Generator 0 -1 -1 0 0 1

1 -1 0 -1 1 0

Translation (0 0) (0 0) (0 0)

A: x + T

B: x + e1 + T

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

B: a/3 + 2b/3 + T

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

-2x = 2a/3 + b/3 + T

With t = b , x = -a/3 -2b/3 = 2a/3 + b/3

Example 2: hyper-quartz

Archetype: 4-dimensional

Tetrahedral coordenation

Cubic-orthogonal family:

(e1-e4), (e2-e5), (e3-e6), ∑ei

Lengths: √2, √2, √2, √6

Space group: 25/08/03/003

1

3

4

6

5

2

b

a

Nets as quotients of the minimal net

100

-1 -1 0

010

N/<111>

N (T = Z3)

001

<111> = translation subgroup generated by 111

N/<111>: T = <{100, 010}>

b

a

(Periodic) voltage net

 = 1 1 1

100

010

Embeddings as projections of the archetype

A

3-d archetype,

2-d structures: 1-d kernel

1

4

2

3-cycle: 3.92, 93

C

6

5

4-cycle: 4.82

D

3

B

Check the space group of 4.82

Kernel: c = e4 + e6 + e3 – e2

A

Generators:

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

(e4,-e6)(e2,e3)(e1,-e1)(A,C)

1

4

2

Cycle basis:

a = e1 + e4 + e5 – e3

b = e1 + e2 - e5 + e6

Notice: a┴b ┴ c

B

6

5

C

D

3

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)

p4mm

Generator 0 -1 0 -1

1 0 -1 0

Translation (0 0) (0 0)

e4 e6 e3 -e2

Walk: A → B → C → D → A

Final translation : c = 0

Find the space group of quartz

A

a = 1000 = e1 - e4

b = 0100 = e2 - e5

c = 0010 = e3 - e6

d = 0001 = e4 + e5 + e6

1

3

4

6

Hyper quartz = N[K3(2)]

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

5

B

C

2

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

b

a

Isomorphism class of a net

b

b

a

Find the labeled quotient graph for the double cell

Isomorphism class of a net

4, a

Kernel: e1-e2

b

b

: (e1, e2)(e3, e4)

3

2

1

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-1o [.(W)] ,

where δ is a path in G from O to (O)

Isomorphism class of a net

t(o. q-1o [W]) = o.q-1o [.(W)]

t: new translation

o: arbitrary origin in N

: boundary operator

q: N  G = N/T

W: walk in G starting at O=q(o)

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

o. q-1o [W]: defines a vertex of N

δ: 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

t(o. q-1o [W]) = o.q-1o [.(W)]

4, a

Kernel: e1-e2

b

b

Extension by  = (e1, e2)(e3, e4)

A

B

3

2

1

a

b

a

b

t:

{e3, e4}

{e1, e2}

b

t ?

t2 = a

{A, B}

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

t 2 (o) = t (o.q-1o [e3]) = o.q-1o [e3.(e3)] = o.q-1o [e3.e4] = a(o)

{2,7}, 100

e1

B

A

e2

{B,C}

{A,D}

{1,8}

e5

e6

e3

e4

Kernel: e2 – e1 = e7 – e8

e3 – e4 = e6 – e5

e7

e8

C

D

Show that cristobalite has the diamond net

{4,5}, k

{3,6}, l

100

010

010

101

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

001

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

Non-crystallographic nets

T

a

σ

b

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

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

e

t

t

σ: (e, -e)

A

B

Non-crystallographic nets

T’

a

σ

b

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

Local automorphisms = T’ x {I, σ}

t’

-t’

tfc / < (001), (110) >

b0

T

c0

A

Local automorphisms = not abelian

But : (b0, c0) ↔ (b, c)

1

1

The 1-periodic graph is a geodesic fiber

B

C

F

01

01

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

A

G

G/T

01

01

at

bt

ct

B

C

10

F/<01> is a

subgraph

of G/T

G 21

21

G 10

10

10

-1 0

-1 0

The 34.6 net (G/T = K6)

10

10

11

10

(Counterclockwise

oriented cycles)

b

a

c

d

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.

disjoint cycles

tangent cycles

2-cycles

AB, CD: 10

10

B

A

10

01

3-cycles

ABC: 00, 01, 11, 10

ABD: 00, 01, -1 0, -1 1

ACD: 00, -1 0, 0 -1, - 1 -1

BCD: 00, 0 -1, 1 -1, 10

01

01

C

D

10

4-cycles

ABCD: 00, 01, 10, 11, 1 -1

ABDC: 01, 11, -1 1

Strong geodesic lines: 10, 01

Geodesic fibers: 11, -1 1

10

B

A

b

a

c

d

C

D

10

G 10

B

A

10

b

a

01

01

01

c

d

C

D

G 11

A

b

a

11

B

c

d

C

D

G 11

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

01

11

10

10

-1 0

0 -1

Graphite layer

0 -1

-1 -1

Def: vector star in the direction of geodesic fibers

Topological density

ρ = lim ∑k=1,r Ck / rn

Ck: kth coordination number

n : dimension of the net

r→∞

ρ = Z.{Σσ f(σ)}/ n!

Z : order of the quotient graph

f(σ) : frequency of the face σ of the cycles figure

= inverse product of the lengths of the vertices

of the face of the cycles figure

Topological density: graphite layer

01

11

f(σ1) = 1/(2.2)

σ1

10

-1 0

ρ = Z.{Σσ f(σ)}/ n!

ρ = 2.6.(2.2)-1/2! = 3/2

0 -1

-1 -1

Topological density: net 32.4.3.4

01

3-cycle

-1 1

11

2-cycle

-1 0

10

ρ = 4.8.(2.3)-1/2! = 8/3

-1 -1

1 -1

0 -1

01

3-cycle

-1 1

11

2-cycle

10

-1 -1

1 -1

0 -1

Find the topological density the β-W net

A

01

01

B

C

10

ρ = 3.8.(2.3)-1/2! = 2

Plane nets

a

A

01

10

b

b

B

C

a

Embedding of the quotient graph in the torus

Face boundaries project on oriented trails of

the quotient graph

Oriented trails are based on oriented edges

A

a

b10

e2, 01

e1, 10

c

c10

B

C

a0-1

a1-1

A

b1-1

e2, 01

e1, 10

B

C

K4 revisited

3.92, 93

4.82

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

a

b

b

a

A plane net from K3, 3

Cyclomatic number:

υ = 9 – 6 + 1 = 4

4.102, 42.10

Kernel : two tangent 4-cycles

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

Some references
• Beukemann A., Klee W.E., Z. Krist. 201, 37-51 (1992)
• Carlucci L., Ciani G., Proserpio D.M., Coord. Chem. Rev. 246, 247-289 (2003)
• Chung S.J., Hahn Th., Klee W.E., Acta Cryst. A40, 42-50 (1984)
• Delgado-Friedrichs O., Lecture Notes Comp. Sci. 2912, 178-189 (2004)
• Delgado-Friedrichs O., O´Keeffe M., Acta Cryst. A59, 351-360 (2003)
• Blatov V.A., Acta. Cryst. A56, 178-188 (2000)
• Eon J.-G., J. Solid State Chem. 138, 55-65 (1998)
• Eon J.-G., J. Solid State Chem. 147, 429-437 (1999)
• Eon J.-G., Acta Cryst.A58, 47-53 (2002)
• Eon J.-G., Acta Cryst, A60, 7-18 (2004)
• Eon J.-G., Acta Cryst A61, 501-511 (2005)
• Harary F., Graph Theory, Addison-Wesley (1972)
• Klee W.E., Cryst. Res. Technol., 39(11), 959-960 (2004)
• Klein H.-J., Mathematical Modeling and Scientific Computing, 6, 325-330 (1996)
• Schwarzenberger R.L.E., N-dimensional crystallography, Pitman, London (1980)
• Gross J.L, Tucker T.W., Topological Graph Theory, Dover (2001)