ECM-23, 4-6 August 2006, Leuven
Download
1 / 120

Jean-Guillaume Eon - UFRJ - PowerPoint PPT Presentation


  • 83 Views
  • Uploaded on

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

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Jean-Guillaume Eon - UFRJ' - jenette-miller


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript

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


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)


P3

C5

B4


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


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


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

G

T

T: spanning tree of G


Hamiltonian graph graph with a spanning cycle
Hamiltonian graph: graph with a spanning cycle main graph

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 edges

  • 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 edges

  • 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 edges

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 edges

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 edges

  • 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 edges

[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)> )


[ edgese3]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 edges

  • 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 edges

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 F edges2 = {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 edges

u

v

Find δ(u + v + w)


w edges

u

v

δu


w edges

u

v


w edges

u

v

δu + δv


w edges

u

v

δu + δv


w edges

u

v


w edges

u

v

δu + δv + δw


w edges

u

v


Complementarity l 1 c c

u edges

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: K edges2(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 edges

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 edges

  • (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 edges

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: edges1. 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 edges

Example 1: the square net

A cycle (with shortcuts)

Sum of three strong rings


Example 2 the cubic net
Example 2: the cubic net edges

A

D

Strong rings:

Cz =ABCDA

Cx = BEFCB

Cy = CFGDC

B

C

G

E

F


= edgesCx + Cy

A cycle that is not a ring:

CBEFGDC

A

D

B

C

CF = shortcut

G

E

F


= C edgesx + 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: edges2.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 edges

Yi

Infinite path (shortcut)

Geodesics

Xj

Strong geodesics


Example 2 the w net
Example 2: the edgesβ-W net

Strong geodesic

Are the following infinite paths geodesics

or strong geodesics?



Geodesic edges


Example 3: the 3 edges4.6 net

Not every periodic graph has strong geodesics!


Geodesic fiber
Geodesic fiber edges

  • 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 w net
Example: the edgesβ-W net


The 3 edges4.6 net


Crystallographic nets
Crystallographic nets edges

  • 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 edges

  • 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 edges

  • 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 edges

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 edges

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 edges

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


Example the w net1
Example: the edgesβ-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


W net quotient graph
β edges-W net: quotient graph

a

c

b


Reo 3 vertex and edge lattices
ReO edges3: vertex and edge lattices


Quotient graph reo 3
Quotient graph: ReO edges3

K1,3(2)



Symmetry
Symmetry edges

  • : 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
ReO edges3: /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 edges

  • 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 edges

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 edges

  • 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 edges

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 edges

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 edges1 -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 edges

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 edges

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 edges

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 edges

a

(Periodic) voltage net

 = 1 1 1

100

010


Embeddings as projections of the archetype
Embeddings as projections of the archetype edges

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 4.8 edges2

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 edges

(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 edges

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 edges

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 edges

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 edges

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 edges

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 edges

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 edges

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 edges

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 edges

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 edges

T’

a

σ

b

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

Local automorphisms = T’ x {I, σ}

t’

-t’


Tfc and derived nets
tfc and derived nets edges

A

100

010

B

C

001


Tfc 001
tfc/<(001)> edges

A

01

10

B

C


Tfc 001 110
tfc / < (001), (110) > edges

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 edges

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 edges 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 edges

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 edges

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


10 edges

B

A

b

a

c

d

C

D

10

G 10


B edges

A

10

b

a

01

01

01

c

d

C

D

G 11


A edges

b

a

11

B

c

d

C

D

G 11


Geodesic fibers and local automorphisms
Geodesic fibers and local automorphisms edges

  • 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 edges

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 edges

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 edges

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 3 edges2.4.3.4

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 edges

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 edges

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 edges 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
K edges4 revisited

3.92, 93

4.82

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


A plane net from k 3 3

a edges

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 edges

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 edges

  • 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)


ad