Jean-Guillaume Eon - UFRJ

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

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

3. 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

4. Crystal structures as topological objects

5. Example: ReO3 1-Complex = Embedding of the Crystallographic net

6. 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

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

8. 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

9. 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

10. 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!

11. 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)

12. P3 C5 B4

13. K4 K3(2) K2, 2, 2

14. 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

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

16. Subgraph of a graph Component = maximum connected subgraph Tree = component of a forest T2 T1 G = T1 T2

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

18. Hamiltonian graph: graph with a spanning cycle Show that K2, 2, 2 is a hamiltonian graph

19. Eulerian graph: graph with a closed trail “covering” all edges Show that K2, 2, 2 is an Eulerian graph

20. 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

21. 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

22. 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)

23. 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

24. 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.

25. 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)> )

26. [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

27. 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)

28. 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

29. 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)

30. w u v Find δ(u + v + w)

31. w u v δu

32. w u v

33. w u v δu + δv

34. w u v δu + δv

35. w u v

36. w u v δu + δv + δw

37. w u v

38. 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*

39. 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

40. 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

41. 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.

42. 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}

43. 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

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

45. Example 2: the cubic net A D Strong rings: Cz =ABCDA Cx = BEFCB Cy = CFGDC B C G E F

46. = Cx + Cy A cycle that is not a ring: CBEFGDC A D B C CF = shortcut G E F

47. = Cx + Cy + Cz A ring that is not a strong ring: ABEFGDA A D B C No shortcut! G E F

48. 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.

49. Example 1: the square net Yi Infinite path (shortcut) Geodesics Xj Strong geodesics

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