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
Crystal structures as
topological objects
Crystal 1-Complex
Net
Chemical Geometric Topological
object
object
object
Atom
Point
Vertex
Bond
Line
Edge
Example: ReO3
1-Complex
= Embedding of the Crystallographic net
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
V = {u, v, w}
e2
e1
u
e3
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
Multiple edges
loop
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
vertices
Incidence relationship
Adjacent
edges
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
Edges only are enough to define the walk!
cycle
Nomenclature
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•
•
•
•
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
(2)
K3
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
T1
T2
G = T1  T2
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
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)
B
A
A---B|=3
|A---B|=4
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’ {f E (e)} = ( f V (u), f V (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
e3
C3
b
f
e2
u
e1
e5
v
e4
f(u) = f(v) = a
f(w) = b
f(e1) = e4
f(e2) = f(e3) = e5
f(vw) = f(v)f(w)
a
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
C3
Generators for Aut(C3)
e2
e1
{
u
e3
w
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.
w
e3
C3
u
e1
[e1]R
qR
e2
v
[u]R
B1 = C3/R
R: generated by (e1, e2, e3)
( noted R = <(e1, e2, e3)> )
x
e4
u
e3
w
C4
e1
[e3]R
e2
qR
[x]R
[u]R
[e2]R
v
[e1]R
R = <(u, v)(x, w)>
Find the quotient graph C4/R
Cycle and cocycle spaces on Z
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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
6
D
C
3
2
Cycle-vector: e1+e2-e3
Cocycle-vector: δA = -e1+e2+e4
5
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)
δu
w
u
v
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
Complementarity: L1 = C  C*
•
•
•
•
Scalar product in L1: ei.ej = ij
dim(C*) = |G|-1
dim(C) + dim(C*) = ||G||
C  C*
C6
u
C6 . u = 0
Example: K2(3)
e1
e2
A
B
e3
L1:
e1 , e2 , e3
C:
e2 – e1 , e3 – e2
C*:
e1 + e2 + e3
CC = M.E
Natural basis (E)
}
Cycle-cocycle basis (CC)
-1 1 0
M = 0 -1 1
1 1 1
Write the cycle-cocycle matrix
for the graph below
A
e1
1
0
M= 0
1
-1
e4
e2 e3
B
e5
C
-1
1
0
1
-1
0
-1
1
1
0
0
0
-1
1
0
0
1
0
0
1
Periodic graphs
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•
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•
•
(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
A
D
Strong rings:
B
C
G
E
F
Cz = ABCDA
Cx = BEFCB
Cy = CFGDC
A cycle that is not a ring:
CBEFGDC = Cx + Cy
A
B
D
C
CF = shortcut
G
E
F
A ring that is not a strong ring:
ABEFGDA = Cx + Cy + Cz
A
B
D
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)
Xj
Geodesics
Strong geodesics
Example 2: the β-W net
Strong geodesic
Are the following infinite paths geodesics
or strong geodesics?
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.
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 t1,0 as Xj
Xj
(i, j)
X0
(0,0)
  fixes X0 and Y0
 fixes the whole net

L(N) = T
Conversely: Let f be any local 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)
at
bt
V = {at , bt , ct}
ct
E = {atbt , atct , btct , atbt+j ,
atct+j , ctbt+i}
Show that β-W is a crystallographic net and
Find its quotient graph
β-W net: quotient graph
a
b
c
ReO3: vertex and edge lattices
O3
O1
e2
O2
Re
e1
O2
O3
O1
Quotient graph: ReO3
O1
K1,3(2)
e2
e3
e6
Re
e4
O2
e1
e5
O3
Simple structure types
Structure
Quotient graph
NaCl
K2(6)
CsCl
K2(8)
ZnS (sphalerite)
K2(4)
SiO2 (quartz)
CaF2 (fluorite)
K3
(2)
K1,2
(4)
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)
O3
O1
e2
O2
Re
e1
O1
O1
O2
O3
e2
e3
e6
Re
e4
O2
e1
e5
O3
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!
Different nets with isomorphic
quotient graphs
C
B
C
D A
A
D
A
D
C
B
B
Vector method: the labeled quotient
graph as a voltage graph
b
b
a
a
B
C
A
C
A(01)
D
B(10)
A
01
A
D
10 D
01
10
C
B
D A
B
C
B
β-W net: Find the labeled quotient graph
A
at
bt
01
01
ct
B
10
t in Z2, i=(1,0), j=(0,1)
V = {at , bt , ct}
E = {atbt , atct , btct , atbt+j , atct+j , ctbt+i}
C
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)
(3)
Example 1: graphite layer - K2
1
A
2
B
3
b
e1
e2
e3
1-chain space: 3-dimensional
Two independent cycles: a = e1-e2, b = e2-e3
One independent cocycle vector: e1+ e2+e3
a
Generators of Aut[K2(3)]:
1
A
B
2
• (e1, e2, e3)
• (ei , -ei)(A, B)
• (e1, -e3)(e2, -e2)(A, B)
3
Linear operations in the cycle space (point symmetry):
(a, b) = (e1 – e2, e2 – e3) → (e2 – e3, e3 – e1) = (b, - a - b)
0
-1
1
-1
In matrix form: γ{(e1, e2, e3)} =
0
1
γ{(e1, -e3)( e2, -e2)} =
p6mm
-1
0
0
-1
γ{(ei, -ei)} =
1
0
Embedding in
3
E
Inversion of the cycle-cocycle Matrix
a = e 1 – e2
b=
e 2 – e3
c = e1 + e2 + e3
e1
2 1 1
e2 = 1/3 -1 1 1
e3
-1 -2 1
a
b =
c
x
1 -1 0
0 1 -1
1 1 1
x
e1
e2
e3
a
b
c
Barycentric embedding: c = 0
e1 = 1/3 (2a + b)
e2 = 1/3 (-a + b)
e3 = 1/3 (-a – 2b)
Atomic positions
p6mm
Generator
Translation
A: x + T
B: x + e1 + T
0 -1
1 -1
(0 0)
-1 0
0 -1
(0 0)
0
1
(0
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
1
0
0)
Example 2: hyper-quartz
1
3
4
6
5
2
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
Nets as quotients of the minimal net
100
b
N (T = Z3)
010
-1 -1 0
001
a
N/<111>
<111> = translation subgroup generated by 111
N/<111>: T = <{100, 010}>
(Periodic) voltage net
=111
010

100
b
a
Embeddings as projections of the
archetype
A
3-d archetype,
2-d structures: 1-d kernel
1
4
6
D
C
3
2
3-cycle: 3.92, 93
4-cycle: 4.82
5
B
A
001
A
Sr[Si2 ]
(I4 32)
100
100
1
^

D
010
010 D
001
B
C
B
C
A
A
2
01
C
(4.8 )
p4mm
2
3
(3.9 ,9 )
p3 m 1
D
10
10 D
01
B
C
B
Check the space group of
Kernel: c = e4 + e6 + e3 – e2
A
1
4
6
C
B
3
2
4.8
Generators:
(e4,e6,e3,-e2)(e1,-e5,-e1,e5)(A,B,C,D)
(e4,-e6)(e2,e3)(e1,-e1)(A,C)
2
5
D
Cycle basis:
a = e 1 + e4 + e5 – e3
b = e 1 + 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)
p4mm
Generator
Translation
0
1
(0
-1
0
0)
0 -1
-1 0
(0 0)
e4 e6
e3 -e2
Walk: A → B → C → D → A
Final translation : c = 0
Find the space group of quartz
A
1
a = 1000 = e1 - e4
b = 0100 = e2 - e5
c = 0010 = e3 - e6
d = 0001 = e4 + e5 + e6
3
4
6
Hyper quartz = N[K3(2)]
5
B
2
C
Quartz = N[K3(2)]/<1110>
Point symmetry
Generators:

(1,4)(2,5)(3,6)

(1,2,3)(4,5,6)(A,B,C)

(1,6)(2,5)(3,4)(B,C)
order 2
order 3
order 2
order 12
(e1 + e2 + e3 - e4 - e5 - e6 = 0 )
Basis: (a, b, d)
: (a, b, d) (-a, -b, d)
: (a, b, d)  (b, -a-b, d)
: (a, b, d)  (-a-b, b, -d)
C2 rotation  d
C6 rotation
C3 rotation  d
C2 rotation  d
Translation part
Associated to the C3 rotation:
e1 e2
e3
Walk: A → B → C → A
Final translation : e1+e2+e3 = d

For  and , t=0 (fixed points) 
31 or 62
P6222
Isomorphism class of a net
b
b
a
a
Find the labeled quotient graph for the double cell
b
Isomorphism class of a net
4, a
b
b
1
Kernel: e1-e2
3
: (e1, e2)(e3, e4)
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-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
A
3
1
{e1, e2}
B
2
t:
{e3, e4}
b
Extension by  = (e1, e2)(e3, e4)
t?
a
b

a
b
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)
Show that cristobalite has the diamond net
e1
A
B
e2
{A,D}
{2,7}, 100
{4,5}, k
{3,6}, l
{B,C}
100
010
C
e5 e6
e3 e4
101
e7 e
{1,8}
010
Kernel: e2 – e1 = e7 – e8
e 3 – e4 = e6 – e5
8
D
(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!
t
e
A
B
t
σ: (e, -e)
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 and derived nets
A
100
B
010
001
C
tfc/<(001)>
A
10
B
01
C
tfc / < (001), (110) >
b0
c0
A
1
T
Local automorphisms = not abelian
But : (b0, c0) ↔ (b, c)
1
The 1-periodic graph is a geodesic fiber
B
C
Projection of a geodesic fiber on the
quotient graph: the β-W net
G
G/T
01
at
bt
01
ct
B
F
A
01
01
10
C
F/<01> is a
subgraph
of G/T
The 34.6 net (G/T = K6)
10
G 21
10
11
21
10
G 10
(Counterclockwise
oriented cycles)
10
10
-1 0
-1 0
a
b
d
c
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.
10
2-cycles
AB, CD: 10
AD, BC: 01
B
A
10
01
01
01
C
D
disjoint cycles
tangent cycles
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
10
Strong geodesic lines: 10, 01
Geodesic fibers: 11, -1 1
4-cycles
ABCD: 00, 01, 10, 11, 1 -1
ABDC: 01, 11, -1 1
ADBC: 1 -1, 10, 11
10
B
A
a
b
d
c
C
D
10
G 10
B
A
a
10
b
01
d
01
01
c
C
D
G 11
A
a
b
d
c
11
B
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
-1 -1
0 -1
Def: vector star in the direction of geodesic fibers
Topological density
ρ = lim ∑k=1,r Ck / rn
r→∞
Ck:
n :
kth coordination number
dimension of the net
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
σ1
10
-1 0
f(σ1) = 1/(2.2)
ρ = Z.{Σσ f(σ)}/ n!
ρ = 2.6.(2.2)-1/2! = 3/2
-1 -1
0 -1
Topological density: net 32.4.3.4
01
-1 1
11
-1 0
3-cycle
10
2-cycle
ρ = 4.8.(2.3)-1/2! = 8/3
-1 -1
1 -1
0 -1
Find the topological density the β-W net
A
01
-1 1
01
B
11
3-cycle
01
10
10
C
-1 -1
1 -1
0 -1
ρ = 3.8.(2.3)-1/2! = 2
2-cycle
Plane nets
a
A
10
B
01
b
C
b
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 b
10
c
c10
a0-1 b
1-1
e2, 01
e1, 10
B
a1-1
C
A
e1, 10
B
e2, 01
C
K4 revisited
3.92, 93
4.82
Two classes of trails: 3-cycles and 4-cycles
A plane net from K3, 3
a
b
b
a
Cyclomatic number:
υ=9–6+1=4
Kernel : two tangent 4-cycles
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
Some references
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•
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•
•
•
•
•
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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)