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Graph theory: fundamentals and applications to crystallographic and crystallochemical problems. The vector method Jean-Guillaume Eon - UFRJ ECM-23, 4-6 August 2006, Leuven

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

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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

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Crystal structures as topological objects

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Example: ReO 3 1-Complex Crystallographic net= Embedding of the

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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 V 2 u v w m (e 1 ) = (u, v) m (e 2 ) = (v, w) m (e 3 ) = (w, u) V = {u, v, w}E = {e 1, e 2, e 3 } e1e1 e2e2 e3e3 e 1 = uv e 1 -1 = vu

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Simple graphs and multigraphs loop Multiple edges Simple graph: graph without loops or multiple edges Multigraph: graph with multiple edges

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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 Adjacent edges Incidence relationship

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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

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a.b.j.i trail a.b.j.d path b.j.d.e.f.g.h.i closed trailb.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

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Nomenclature P n path of n edges C n cycle of n edges B n bouquet of n loops K n complete graph of n vertices K n (m) complete multigraph of n vertices with all edges of multiplicity m K n1, n2,... nr complete r-partite graph with r sets of n i vertices (i=1,..r)

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C5C5 P3P3 B4B4

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K 3 (2) K4K4 K 2, 2, 2

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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

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Determine κ(G) and λ(G) G λ(G) = 3 κ(G) = 2

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Subgraph of a graph Component = maximum connected subgraph Tree = component of a forest T1T1 T2T2 G = T 1 T 2

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Spanning graph = subgraph with the same vertex set as the main graph G T: spanning tree of G T

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Hamiltonian graph: graph with a spanning cycle Show that K 2, 2, 2 is a hamiltonian graph

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Eulerian graph: graph with a closed trail “covering” all edges Show that K 2, 2, 2 is an Eulerian graph

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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|=4 d(A, B)=3

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Morphisms of graphs A morphism between two graphs G(V, E, m) and G’(V’, E’ m’) is a pair of maps f V and f E 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

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e3e3 e1e1 e2e2 v w C3C3 f u b a e4e4 e5e5 f(u) = f(v) = a f(w) = b f(e 1 ) = e 4 f(e 2 ) = f(e 3 ) = e 5 Example: an onto morphism in non-oriented graphs f(vw) = f(v)f(w)

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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 e1e1 e2e2 e3e3 C3C3 Generators for Aut(C 3 ) order 6 (e 1, e 2, e 3 ) order 3 (e 1, e 3 -1 ) (e 2, e 2 -1 ) order 2 {

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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: q R (x) = [x] R (for x in V or E), defines a graph homomorphism.

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e3e3 e1e1 e2e2 u v w R: generated by (e 1, e 2, e 3 ) ( noted R = ) C3C3 [u]R[u]R [e1]R[e1]R B 1 = C 3 /R qRqR

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uv wx e1e1 e2e2 e3e3 e4e4 R = Find the quotient graph C 4 /R C4C4 [u]R[u]R [x]R[x]R [e2]R[e2]R [e3]R[e3]R [e1]R[e1]R qRqR

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Cycle and cocycle spaces on Z 0-chains: L 0 ={∑λ i u i for u i V and λ i in Z} 1-chains: L 1 ={∑λ i e i for e i E and λ i in Z} Boundary operator: ∂e = v – u for e = uv Coboundary operator: δu = ∑e i for all e i = uv i 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)

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Cycle and cocycle vectors A B C D Cycle-vector: e 1 +e 2 -e 3 Cocycle-vector: δA = -e 1 +e 2 +e 4 ∂e 1 = A - D

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Cycle and cut spaces on F 2 = {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)

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u v w Find δ(u + v + w)

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u v w δuδu

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u v w

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u v w δu + δv

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u v w

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u v w

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u v w δu + δv + δw

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u v w

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Complementarity: L 1 = C C* Scalar product in L 1 : e i.e j = ij dim(C*) = |G|-1 dim(C) + dim(C*) = ||G|| C C* u C6C6 C 6. u = 0

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Example: K 2 (3) A B e1e1 e2e2 e3e3 L 1 : e 1, e 2, e 3 Natural basis (E) C: e 2 – e 1, e 3 – e 2 C*: e 1 + e 2 + e 3 } Cycle-cocycle basis (CC) CC = M.E M =

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Write the cycle-cocycle matrix for the graph below A BC e1e1 e2e2 e3e3 e4e4 e5e5 M =

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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.

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Example: the square net V = Z 2 E = {pq | q-p = ± a, ± b} a = (1,0), b = (0,1) T = {t r : t r (p) = p+r, r in Z 2 }

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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

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Example 1: the square net A cycle (with shortcuts) Sum of three strong rings A strong ring

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Example 2: the cubic net Strong rings: C z = ABCDA C x = BEFCB C y = CFGDC A BC D EF G

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A cycle that is not a ring: CBEFGDC A BC D EF G CF = shortcut = C x + C y

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A ring that is not a strong ring: ABEFGDA No shortcut! A B D EF G C = C x + C y + C z

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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.

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Example 1: the square net Strong geodesics Geodesics YiYi XjXj Infinite path (shortcut)

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Are the following infinite paths geodesics or strong geodesics? Strong geodesic Example 2: the β-W net

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Strong geodesic

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Geodesic

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Example 3: the net Not every periodic graph has strong geodesics!

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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.

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Example: the β-W net

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The net

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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.

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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

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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.

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(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 t i, j -1. f = 1 T={t i,j : t i,j (m, n) = (m+i, n+j)} : T L(N) Y0Y0 YiYi X0X0 XjXj Y 0 invariant by t 0,1 as Y i X 0 invariant by t 1,0 as X j u fixes X 0 and Y 0 u fixes the whole net u L(N) = T

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Example: the square net Simple, Every automorphism of the infinite graph can be associated to an isometry of the plane Crystallographic net 4-connected L(N) = T free abelian One vertex orbit Two edge orbits

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Quotient graph of a periodic graph: the square net Q = G/T, T translation subgroup T of the automorphism group

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Example: the β-W net Show that β-W is a crystallographic net and Find its quotient graph atat btbt ctct t in Z 2, i=(1,0), j=(0,1) V = {a t, b t, c t } E = {a t b t, a t c t, b t c t, a t b t+j, a t c t+j, c t b t+i }

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β-W net: quotient graph a b c

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ReO 3 : vertex and edge lattices

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Quotient graph: ReO 3 K 1,3 (2)

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Simple structure types

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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)

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ReO 3 : /T Aut(G) m3m C 4 Rotation: (e 1, e 3, e 2, e 4 ) C 3 Rotation: (e 1, e 3, e 5 ) (e 2, e 4, e 6 ) Mirror: (e 1, e 2 )

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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!

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Different nets with isomorphic quotient graphs

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Vector method: the labeled quotient graph as a voltage graph

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β-W net: Find the labeled quotient graph atat btbt ctct t in Z 2, i=(1,0), j=(0,1) V = {a t, b t, c t } E = {a t b t, a t c t, b t c t, a t b t+j, a t c t+j, c t b t+i } A BC 10 01

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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)

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Example 1: graphite layer - K 2 (3) AB 1-chain space: 3-dimensional Two independent cycles: a = e 1 -e 2, b = e 2 -e 3 One independent cocycle vector: e 1 + e 2 +e 3 e1e1 e2e2 e3e3 a b

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1 2 3 AB Generators of Aut[K 2 (3) ]: (e 1, e 2, e 3 ) (e i, -e i )(A, B) (e 1, -e 3 )(e 2, -e 2 )(A, B) Linear operations in the cycle space (point symmetry): (a, b) = (e 1 – e 2, e 2 – e 3 ) → (e 2 – e 3, e 3 – e 1 ) = (b, - a - b) In matrix form: γ{(e 1, e 2, e 3 )} = γ{(e 1, -e 3 )( e 2, -e 2 )} = γ{(e i, -e i )} = p6mm

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Embedding in E 3 Inversion of the cycle-cocycle Matrix a = e 1 – e 2 b = e 2 – e 3 c = e 1 + e 2 + e 3 a e 1 b = x e 2 c e 3 e a e 2 = 1/ x b e c Barycentric embedding: c = 0 e 1 = 1/3 (2a + b) e 2 = 1/3 (-a + b) e 3 = 1/3 (-a – 2b)

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Atomic positions p6mm Generator Translation (0 0) (0 0) (0 0) A: x + T B: x + e 1 + 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 A: 2a/3 + b/3 + T B: a/3 + 2b/3 + T

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Example 2: hyper-quartz Archetype: 4-dimensional Tetrahedral coordenation Cubic-orthogonal family: (e 1 -e 4 ), (e 2 -e 5 ), (e 3 -e 6 ), ∑e i Lengths: √2, √2, √2, √6 Space group: 25/08/03/003

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Nets as quotients of the minimal net = translation subgroup generated by 111 N/ : T = N (T = Z 3 ) N/ ab

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(Periodic) voltage net ab =

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Embeddings as projections of the archetype A B C D d archetype, 2-d structures: 1-d kernel 3-cycle: 3.9 2, cycle: 4.8 2

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Check the space group of A B DC Kernel: c = e 4 + e 6 + e 3 – e 2 Generators: (e 4,e 6,e 3,-e 2 )(e 1,-e 5,-e 1,e 5 )(A,B,C,D) (e 4,-e 6 )(e 2,e 3 )(e 1,-e 1 )(A,C) Cycle basis: a = e 1 + e 4 + e 5 – e 3 b = e 1 + e 2 - e 5 + e 6 Notice: a ┴ b ┴ c

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Point symmetry (a, b) → (-e 5 + e 6 + e 1 + e 2, -e 5 – e 4 – e 1 + e 3 ) = (b, -a) (a, b) → (-e 1 – e 6 + e 5 – e 2, -e 1 + e 3 - e 5 – e 4 ) = (-b, -a) p4mm Generator Translation (0 0) (0 0) Walk: A → B → C → D → A Final translation : c = 0 e 4 e 6 e 3 -e 2

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Find the space group of quartz a = 1000 = e 1 - e 4 b = 0100 = e 2 - e 5 c = 0010 = e 3 - e 6 d = 0001 = e 4 + e 5 + e 6 Hyper quartz = N[K 3 (2) ] Quartz = N[K 3 (2) ]/ A B C

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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 (e 1 + e 2 + e 3 - e 4 - e 5 - e 6 = 0 ) Basis: (a, b, d) order 12 : (a, b, d) (-a, -b, d) C 2 rotation d : (a, b, d) (b, -a-b, d)C 3 rotation d : (a, b, d) (-a-b, b, -d)C 2 rotation d C 6 rotation

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Translation part Associated to the C 3 rotation: Walk: A → B → C → A Final translation : e 1 +e 2 +e 3 = d 3 1 or 6 2 e 1 e 2 e 3 For and , t=0 (fixed points) P6 2 22

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Isomorphism class of a net a b Find the labeled quotient graph for the double cell a b b

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Isomorphism class of a net 4, a b b 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 -1 o [W]) = o. q -1 o [ . (W)], where δ is a path in G from O to (O) Kernel: e 1 -e 2 : (e 1, e 2 )(e 3, e 4 )

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Isomorphism class of a net t(o. q -1 o [W]) = o. q -1 o [ . (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 -1 o [W]: walk in N starting at o and projecting on W o. q -1 o [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 -1 o [ ]

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Isomorphism class of a net 4, a b b Kernel: e 1 -e 2 Extension by = (e 1, e 2 )(e 3, e 4 ) b AB {A, B} {e 1, e 2 } {e 3, e 4 } With o in q -1 (A) and = e 3 : t (o) = o. q -1 o [e 3 ] t 2 (o) = t (o. q -1 o [e 3 ]) = o. q -1 o [e 3. (e 3 )] = o. q -1 o [e 3.e 4 ] = a(o) t 2 = a t ? abab abab t:t: t(o. q -1 o [W]) = o. q -1 o [ . (W)]

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Show that cristobalite has the diamond net e1e1 e8e8 e7e7 e6e6 e5e5 e4e4 e3e3 e2e2 A C B D Kernel: e 2 – e 1 = e 7 – e 8 e 3 – e 4 = e 6 – e 5 (1,8)(2,7)(3,6)(4,5)(A,D)(B,C) {A,D} {B,C} {1,8} {2,7}, 100 {4,5}, k e 8 – e 4 + e 1 – e 5 = 001 → e 1 – e 5 + e 8 – e 4 = 001 2k = e 4 – e 8 + e 5 – e 1 = {3,6}-{4,5} = 010 l = k {3,6}, l

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Non-crystallographic nets Local automorphisms = T x {I, σ} : abelian but not free! T σ T = a b tt e AB σ: (e, -e)

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Non-crystallographic nets Local automorphisms = T’ x {I, σ} T’ σ T’ = a b t’ -t’

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tfc and derived nets A BC

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A BC tfc/

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A B C 1 1 Local automorphisms = not abelian But : (b 0, c 0 ) ↔ (b, c) b0b0 c0c0 T The 1-periodic graph is a geodesic fiber

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Projection of a geodesic fiber on the quotient graph: the β-W net atat btbt ctct A BC F G G/T F/ is a subgraph of G/T 01

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The net (G/T = K 6 ) (Counterclockwise oriented cycles) 21 G 21 G

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Determine the quotient graph G/T of the net, list all the cycles of G/T; find the strong geodesic lines of and its geodesic fibers. a d b c

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A D B C cycles AB, CD: 10 AD, BC: 01 3-cycles ABC: 00, 01, 11, 10 ABD: 00, 01, -1 0, -1 1 ACD: 00, -1 0, 0 -1, BCD: 00, 0 -1, 1 -1, 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 disjoint cycles tangent cycles Geodesic fibers: 11, -1 1

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a d b c A D B C 10 G 10

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A D B C G a d b c

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A D B C 11 G 11 a d b c

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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.

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Cycle figure Def: vector star in the direction of geodesic fibers Graphite layer

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Topological density ρ = lim ∑ k=1,r C k / r n C k : k th coordination number n : dimension of the net r→∞ ρ = Z.{Σ σ 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

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Topological density: graphite layer σ1σ1 f(σ 1 ) = 1/(2.2) ρ = Z.{Σ σ f(σ)}/ n! ρ = 2.6.(2.2) -1 /2! = 3/2

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Topological density: net cycle 2-cycle ρ = 4.8.(2.3) -1 /2! = 8/3

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Find the topological density the β-W net BC 01 A cycle 2-cycle ρ = 3.8.(2.3) -1 /2! = 2

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Plane nets Embedding of the quotient graph in the torus A BC a a bb Face boundaries project on oriented trails of the quotient graph

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Oriented trails are based on oriented edges e 1, 10 e 2, 01 A BC a b 10 c a 0-1 b 1-1 a 1-1 c 10 e 1, 10 e 2, 01 A BC

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K 4 revisited 3.9 2, Two classes of trails: 3-cycles and 4-cycles

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A plane net from K 3, 3 Cyclomatic number: υ = 9 – = 4 Kernel : two tangent 4-cycles a a b b ,

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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

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Non-ambiently isotopic embeddings of the same net ThSi 2 SrSi 2

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Self-interpenetrated ThSi 2

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Self-interpenetrated SrSi 2

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Some references Beukemann A., Klee W.E., Z. Krist. 201, (1992) Carlucci L., Ciani G., Proserpio D.M., Coord. Chem. Rev. 246, (2003) Chung S.J., Hahn Th., Klee W.E., Acta Cryst. A40, (1984) Delgado-Friedrichs O., Lecture Notes Comp. Sci. 2912, (2004) Delgado-Friedrichs O., O´Keeffe M., Acta Cryst. A59, (2003) Blatov V.A., Acta. Cryst. A56, (2000) Eon J.-G., J. Solid State Chem. 138, (1998) Eon J.-G., J. Solid State Chem. 147, (1999) Eon J.-G., Acta Cryst. A58, (2002) Eon J.-G., Acta Cryst, A60, 7-18 (2004) Eon J.-G., Acta Cryst A61, (2005) Harary F., Graph Theory, Addison-Wesley (1972) Klee W.E., Cryst. Res. Technol., 39(11), (2004) Klein H.-J., Mathematical Modeling and Scientific Computing, 6, (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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