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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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slide1

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
Summary
  • Motivation
  • Fundamentals of graph theory
  • Crystallographic nets and their quotient graphs
  • Space group and isomorphism class
  • Topological and geometric properties
objectives
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 reo 3
Example: ReO3

1-Complex

= Embedding of the

Crystallographic net

graph theory harary 1972

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
Simple graphs and multigraphs

loop

Multiple 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

Adjacent

edges

Adjacent

vertices

Incidence relationship

Some elementary definitions
walks paths and cycles
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
slide10

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
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)
slide12

P3

C5

B4

slide13

K4

K3(2)

K2, 2, 2

connectivity
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
slide15

G

κ(G) = 2

λ(G) = 3

Determine κ(G) and λ(G)

slide16

Subgraph of a graph

Component = maximum connected subgraph

Tree = component of a forest

T2

T1

G = T1 T2

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

Show that K2, 2, 2 is an Eulerian graph

distances in a 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
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
slide22

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)

aut g group of automorphisms

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
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.
slide25

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)> )

slide26

[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
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
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 f 2 0 1 non oriented graphs
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)
slide30

w

u

v

Find δ(u + v + w)

slide31

w

u

v

δu

slide32

w

u

v

slide33

w

u

v

δu + δv

slide34

w

u

v

δu + δv

slide35

w

u

v

slide36

w

u

v

δu + δv + δw

slide37

w

u

v

complementarity l 1 c c

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 k 2 3
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

write the cycle cocycle matrix for the graph below

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
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
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
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 strong ring

Example 1: the square net

A cycle (with shortcuts)

Sum of three strong rings

example 2 the cubic net
Example 2: the cubic net

A

D

Strong rings:

Cz =ABCDA

Cx = BEFCB

Cy = CFGDC

B

C

G

E

F

slide46

= Cx + Cy

A cycle that is not a ring:

CBEFGDC

A

D

B

C

CF = shortcut

G

E

F

slide47

= 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
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 net1
Example 1: the square net

Yi

Infinite path (shortcut)

Geodesics

Xj

Strong geodesics

example 2 the w net
Example 2: the β-W net

Strong geodesic

Are the following infinite paths geodesics

or strong geodesics?

slide53

Example 3: the 34.6 net

Not every periodic graph has strong geodesics!

geodesic fiber
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 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
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
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 net1
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 net2
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
Quotient graph of a periodic graph: the square net

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

example the w net1
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
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)
reo 3 t aut g m 3 m
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
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!
slide73

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
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 k 2 3

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

slide76

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

embedding in e 3

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
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
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

nets as quotients of the minimal net

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}>

periodic voltage net

b

a

(Periodic) voltage net

 = 1 1 1

100

010

embeddings as projections of the archetype
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 8 2
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
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
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 symmetry1
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
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

b

a

Isomorphism class of a net

b

b

a

Find the labeled quotient graph for the double cell

isomorphism class of a net1
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)

slide91

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 net2
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)

slide93

{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
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 nets1
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
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

slide99

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

the 3 4 6 net g t k 6

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)

slide101

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.

slide102

disjoint cycles

tangent cycles

2-cycles

AB, CD: 10

AD, BC: 01

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

ADBC: 1 -1, 10, 11

Strong geodesic lines: 10, 01

Geodesic fibers: 11, -1 1

slide103

10

B

A

b

a

c

d

C

D

10

G 10

slide104

B

A

10

b

a

01

01

01

c

d

C

D

G 11

slide105

A

b

a

11

B

c

d

C

D

G 11

geodesic fibers and local automorphisms
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
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
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
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 3 2 4 3 4
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

find the topological density the w net

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
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
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

k 4 revisited
K4 revisited

3.92, 93

4.82

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

a plane net from k 3 3

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

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

Topological invariants of nets in projection on the quotient graph
some references
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)
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