Graph Theory (The Vector Method)

Topology of Crystal Structures: Applications of Graph Theory (the vector method)

Institute of Chemistry, Federal University of Rio de Janeiro, Rio de Janeiro - Brazil

Manila, Workshop on Modern Trends in Mathematical Crystallography, May 20 2017 Outline Part 1: basic concepts Part 2: advanced topics

• Some graph theory • Minimal nets • Underlying nets and topology • Rings and generators • Labelled quotient graphs • Nets as projections • Symmetry • Non-crystallographic nets • Crystallographic nets • Building units • Barycentric embeddings • Groupoids Topology of Crystal Structures

Basic concepts Crystallography and crystallochemistry

(NH4)3PMo12O40

Atomic positions Chemical bonds Topology Electron density maps

Crystal structure and electron density in the apatite-type ionic conductor

La9.71(Si5.81Mg0.18)O26.37

Roushown Ali, Masatomo Yashima, Yoshitaka Matsushita, Hideki Yoshioka, Fujio Izumi −6 −3 Electron density isosurface (0.5 × 10 e·pm ) Journal of Solid State Chemistry 182 (2009) 2846–2851 4+ − in Ca24Al28O64] ·4e (maximum entropy method analysis) superposed on the atomic structure of the unit cell

K. Hayashi, P.V. Sushko, Y. Hashimoto, A.L. Shluger, H. Hosono Nature Communications 5 (2014) 3515 periodic nets and quotient graphs

010 100 Quotient

001

Unit cell:

One Rhenium atom Three oxygen atoms six Re – O bonds Chung S.J., Hahn Th., Klee W.E. Acta Cryst. A40, 42-50 (1984) Combinatorial and Embedding Topology

Strong distortion Hopf link Graph theory (Harary 1972)

A graph G(V, E, i) is defined by: – a set V of vertices, – a set E of edges, – an incidence function i from E to the set of couples {x, y} of elements of V.

V = {u, v, w} E = {e1, e2, e3} e e1 2 i (e1) = {u, v} u i (e2) = {v, w}

i (e3) = {w, u} e3 w Graphs with an orientation (σ)

• For G = (V, E, i), let Gσ = (V, Eσ, iσ) such that: • σ : Eσ  E is a two-to-one, onto mapping • Preimages of e = {u, v} are noted e+and e- • iσ(e+) = (u, v) and iσ(e-) = (v, u)

• Notations: e+ = uv, e- = vu v

V = {u, v, w} E = {e1, e2, e3} G e e1 2 i (e1) = {u, v}

u i (e2) = {v, w}

i (e3) = {w, u} e3 w

v σ + - - - - G V = {u, v, w} E = {e1, e2, e3} E = {e1 , e2 , e3 } e e1 2 i (e1) = (u, v) + e1 = e1 = uv u i (e2) = (v, w) - e1 = vu i (e3) = (w, u) e3 w Graphs and multigraphs

Multiple edges loop

Simple graph: graph without loops or multiple edges Subgraphs

H  G Induced subgraph

V(H)  V(G) V(H) = S  V(G) E(H)  E(G) E(H) maximal in E(G) iH is a restriction of iG Order, size and degrees

• Order of G: |G| = number of vertices of G • Size of G: ||G|| = number of edges of G • Degree of a vertex u: d(u) = number of edges incident to u (loops are counted twice) • Regular graph of degree r: d(u) = r Walks, paths and cycles

• Walk: alternate sequence of vertices and edges mutually adjacent • Closed walk: the last vertex is equal to the first = the last edge is adjacent to the first one • Path: a walk which traverses only once each vertex • Cycle: a closed path • Forest: a graph without cycle a.b.j.i walk b.j.d.e.f.g.h.i closed walk a.b.c path b.j.i cycle

Edges only are enough to define the walk! Nomenclature

• Pn path of n edges

• Cn cycle of n edges

• Bn bouquet of n loops

• Kn complete graph of n vertices {m} • Kn 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

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 points that must be withdrawn to get a disconnected graph

• Line-connectivity: λ(G), minimum number of lines that must be withdrawn to get a disconnected graph G

κ(G) = 2 λ(G) = 3

Determine κ(G) and λ(G). Component = maximum connected subgraph

Tree = component of a forest

T1 T2

F = T1 U T2 Spanning graphs

T: spanning tree of G Morphisms of (oriented) graphs

• A morphism between two graphs G(V, E, i)

and G’(V’, E’ i’) is a pair of maps fV and fE between the vertex and edge sets that respect the incidence relationships:

for i (e) = (u, v) : i’ {f E (e)} = ( f V (u), f V (v)) or: f(uv) = f(u)f(v) • Isomorphism of graphs: 1-1 morphism 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 edges (invariant by reversal of the signs!)

Generators for Aut(C3) C3

e2 e1 (e1, e2, e3)

u (e , e -) (e , e -) { 1 3 2 2 e3 w Special case of the loops

• Inversion of a loop: (e+, e-) • 4th order permutation of two loops: + + - - (e1 , e2 , e1 , e2 )

e2 e1 {3} Reversal of the signs: K2

+ e1 - e1

- e2 - e3

+ + - - Forbidden permutation: (e1 , e2 )( e1 , e3 )

+ + - - Permitted permutation: (e1 , e2 )( e1 , e2 ) Quotient graphs

• Let F be a subgroup of Aut(G) that acts freely on G, • Denote [x] the orbit of x in V or E by F in G. • The quotient G/F is the “graph of the orbits”: V(G/F) = {[v] for v in V} E(G/F) = {[e] for e in E} [e] = [u][v] for e = uv • The graph homomorphism q : q(x) = [x] is called the natural projection of G on G/F w [e1]R q e R 3 e2 C3 u v [u]R e1

B1 = C3/R

R: generated by (e1, e2, e3)

( noted R = <(e1, e2, e3)> ) [e ] e 3 R x 3 w q R [x]R e e 4 C4 2 [e2]R [u]R u v e1 [e ] R = <(u, v)(x, w)> 1 R S = <(u, v, w, x)> T = <(v, x)>

Find the quotient graphs C4/R, C4/S and C4/T

Compare degrees in C4 and its quotients Free action automorphisms

f in Aut(G) acts freely on the graph G if there is no fixed element in V or E by f

(U,V,W) and (A,B,C,D) act freely (V,W), (B,D) and (A,D)(B,C) do not Periodic nets

• A net is a simple, 3-connected graph, which is locally finite (WE Klee, 2004). • (N, T) is a p-periodic net if: – N is a net, – T < Aut(N), is free abelian of rank p, – The number of (vertex and edge) orbits in N by T is finite.

Then, T acts freely on N 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 subgroup T of rank n which is normal in Γ, has finite factor group Γ/T and whose centralizer coincides with T (T is maximal abelian).  Example: the square net

V = Z2 E = {pq | q-p = ± a, ± b} a = (1,0), b = (0,1)

2 T = {tr: tr (p) = p+r, r in Z } Natural projection : examples

<(U,V,W)> C3

Square T net ReO3: vertex and edge lattices

O2 e2 O1 Re e1 O1 O2 O3 ReO3 : quotient graph

K (2)(2) K1,31,3

e2 e1

e e 3 Re 6

e4 e5

O2 O3 The β-W net

At A

Bt Ct

K (2)(2) K33 B C Simple structure types

Structure Quotient graph {6} NaCl K2 {8} CsCl K2 {4} ZnS (sphalerite) K2 {3} SiO2 (quartz) K3 {4} CaF2 (fluorite) K1,2 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 (non-crystallographic nets for different choice of the translation group) • Different 1-complexes can have both isomorphic nets and isomorphic quotient graph (non-ambiently isotopic embeddings) Non-isomorphic nets with isomorphic quotient graphs

C B B A D A C D A

C B Vector method: the labelled quotient graph as a voltage graph b b

a C a B B A A A(01) D C D B(10)

A A

01 D 10 D 10 01 C B C B Labelled Quotient Graphs of Periodic Nets

(N, T) : periodic net

A net N A translation group T

N/T : Labelled Quotient Graph

Two vertex-lattices Three edge-lattices Label vectors in T The kagome net (kgm)

A unit cell in an embedding of kgm and a fundamental transversal with its boundaries.

The fundamental transversal and

the labelled quotient graph obtained after closing half-edges. β-W net: Find the labelled quotient graph

a t 01 01

bt ct

b 10 c

t in Z2, i=(1,0), j=(0,1)

V = {at , bt , ct}

E = {atbt , atct , btct , atbt+j , atct+j , ctbt+i} “Voltage” or “edge-coloured” graphs

• Graphs with an orientation : for e = uv define e- = vu (and e+ = e) • A voltage  of a graph G by a group A is a mapping  : E A with (e-) = [(e)]-1 • Derived graph G : V(G) = {(v, a) in V  A} E(G) = {(e, a) in E  A} (e, a) = (u, a)(v, ab) for e = uv and (e) = b • A acts freely on G : f(x, a) = (x, fa) for x in V or E and f in A Nets and topology: why 3-connectivity? Derived graphs : examples

A =

A = Z2

10 Cycle and cocycle spaces on Z

• 0-chains: ∑λiui for ui  V λi in Z • 1-chains: ∑λiei for ei  E • Boundary operator: ∂e = v – u for e = uv

• Coboundary operator: δu = ∑ei for ei = uvi • Cycle space = Ker(∂) (cycle-vector C : ∂C = 0) • Cocycle (cut) space = Im(δ) (cocycle-vector k = δu) • dim(cycle space) = #E - #V +1 (cyclomatic number) Cycle and cocycle vectors

∂e1 = A - D 1 4 2 Cycle-vector: e1+e2-e3

C Cocycle-vector: δA = -e1+e2+e4 6 5

D 3 B

Find a basis for: 1) the cycle-space and 2) the cocycle-space

of the cartesian product K2 x K3 (trigonal prism) 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) δu + δv + δw δu u w δu + δv

Find δ(u + v + w) Minimal nets

• Cyclomatic number g = number of independent cycles of G = number of chords of a spanning tree T • Choose A = Zg free abelian group of order g • Voltage assignment :  for each chord e of T, (e) is a different generator of Zg  for e in T, (e) = 0 • Minimal net M[G] = derived net G • The translation subgroup of M[G] ≡ Cycle-space of G (3) Example 1: graphite (K2 )

A B 2 b e1 a

e2 3

e3 1-chain space: 3-dimensional

Two independent cycles: a = e1-e2, b = e2-e3

One independent cocycle vector: e1+ e2+e3 Sr[Si]2 = srs : K4

a c

Edge space: 6-dimensional

Three independent cycles: a , b , c Three independent cut vectors : coboundary of three vertices {2} Hyperquartz : K3 A

Archetype: 4-dimensional Tetrahedral coordination 1 3 Cubic-orthogonal family: 4 6 (e1-e4), (e2-e5), (e3-e6), ∑ei 5 Lengths: √2, √2, √2, √6 Space group: 25/08/03/003

B 2 C Edge space: 6-dimensional

Four independent cycles: e1-e4 , e2-e5 , e3-e6 , e1+e2+e3 Two independent cut vectors 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) Symmetry of minimal nets

• 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) Voltages, permutations and crystal symmetry

b -a -b a

C4 = (a, b, -a, -b) b a C2 = (a, -a)(b, -b) Reflections

b b a a

σ = (a, -a) Reflections

b b a a

σ = (b, -b) Reflections

b b a a

σ = (a, b)(-a, -b) Reflections

b b a a

σ = (a, -b)(b, -a) ReO3: /T  Aut(G)  Oh

• C4 Rotation: (e1, e3, e2, e4)

• C3 Rotation: (e1, e3, e5) (e2, e4, e6) O3 • Reflection: (e1, e2) O2 e2 O1 O1 Re e1 O1 O2 O3

e2 e1

e e 3 Re 6

e4 e5

O2 O3 Geometric crystal class: minimal nets

• Generators and conjugacy classes in Aut(G) as edge permutations • Linear operations in the cycle-space • Matrix invariants: order, trace, determinant {3} 1 Generators of Aut(K2 ):

A B • (e , e , e ) 2 1 2 3 • (ei , -ei)(A, B)

3 • (e1, -e3)(e2, -e2)(A, B)

Linear operations in the cycle space (point symmetry):

(a, b) = (e1 – e2, e2 – e3) → (e2 – e3, e3 – e1) = (b, - a - b)

0 -1 p6mm In matrix form: γ{(e1, e2, e3)} = 1 -1

0 1 -1 0

γ{(e1, -e3)( e2, -e2)} = γ{(ei, -ei)} = 1 0 0 -1 Geometric crystal class

When the net is not a minimal net:

The point group of the net is the subgroup of Aut(G) which respects the subspace of the cycle-space with zero net voltage. 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)

1 4 2 (e4,-e6)(e2,e3)(e1,-e1)(A,C)

B Cycle basis: 6 5 a = e1 + e4 + e5 – e3

C 3 D b = e1 + 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 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

a = 1000 = e1 - e4

b = 0100 = e2 - e5

c = 0010 = e3 - e6 1 3 d = 0001 = e4 + e5 + e6 4 6

(2) 5 Hyper quartz = N[K3 ]

Quartz = N[K (2)]/<1110> B 3 2 C Isomorphism class of a net

j i

j i j

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

4 i Kernel: e1-e2 j j Extension: (e1, e2)(e3, e4) 1 3 2

Automorphisms  of the quotient graph G that leave the quotient group of the cycle space by its kernel invariant can be used to extend the translation group if the generated group acts freely: Isomorphism class of a net

a e1 e b e4 b 2 Kernel: e1-e2 e A 3 B Extension: c = (e1, e2)(e3, e4)(A, B)

{e1, e2} {e3, e4} b c c2 = a

{A, B} Show that cristobalite has the diamond net.

e {2,7}, 100 A 1 B e {4,5}, k 2 {A,D} {B,C} 100 {1,8} e e 5 6 010 e3 e4 010 Kernel: e – e = e – e 101 2 1 7 8 e3 – e4 = e6 – e5 e7 e8 C D 001 (1,8)(2,7)(3,6)(4,5)(A,D)(B,C)

e8 – e4 + e1 – e5 = 001 → e1 – e5 + e8 – e4 = 001

2k = e4 – e8 + e5 – e1 = 0 0 -1 {3,6}(-{4,5}) = 010: voltage for {3,6} = 010 + k Representation and embedding

• Euclidian representation, p : V  E  En for u  V, p(u) is a point of En if e = uv, p(e) is the line p(u)p(v) We note p(e) the vector p(v) - p(u) in Rn

• Embedding: p is injective on V p(e)  p(e´) =  unless e ~ e´ K4

Representation with crossing in E2 Embedding in E2 Minimal net: integral embedding

• m = |E(N/T)| • Embedding (p) in the Euclidian space Em with

orthonormal basis {ei, i=1,...m}

• For ei  E(N/T) set p(ei) = ei Fix an origin in the vertex set: p(u) = O • For v  V(N), let w be a walk from u to v

with projection in N/T: q(w) =  xiei

• Set p(v) = O +  xiei Euclidian representations

• Barycentric representation: –  : V(N) → E (Euclidian space)

– r(u)(u) = v  u(v) r(u) = degree of u • Periodic representation: – * : T → T* (translation group in E) – [t(u)] = t*[(u)] for any t  T, u  V(N) • Determination from the labelled quotient graph: - the vertex method - the edge method Barycentric representation: the vertex method

1 0

A B Neighbours of A 00 : B00 , B10 and B0-1

0 -1 e2 b Barycentric equation: 3 (A 00 ) = (B 00 ) + (B 10 ) + (B 0-1 ) e3

3 (A 00 ) = (B00 ) + [(B00 ) + 10] + [(B00 ) + 0-1] e1 (A ) = 0  (B ) = 00 00 a Vertex method and Laplacian matrix

Lii = - d(Vi) L = (Lij)

Lij = m if ViVj  E(G) with multiplivity m, for i  j

(Vi) = Sum of voltages over ingoing edges at Vi

(V): column vector of Vi L.(V) = (a(Vi)) det(L) = 0 ! ((V)): column vector of (Vi)

Solved by eliminating the last row of L and imposing V1 = 0 Example

Draw a barycentric embedding of the 2- 10 01 periodic net derived from the following labelled quotient graph. B C Example A -4 2 2 A -1 -1 10 01 L = 2 -3 1 (V) = B ((V)) = 1 0 2 1 -3 C 0 1 B C C B

Set A at the origin: A -1 B 2 2 -1 -1 -3/8 -1/8 a b = = C -3 1 1 0 -1/8 -3/8 Periodic embeddings: principles

Edge-space = Cycle-space  Cut-space

G: n vertices, m edges

E: m x 1 column vector (edges) B: m x 1 column vector (cycle and cut bases) K: m x m matrix expressing the vectors of B as combinations of edges

B = K.E  E = K-1.B Barycentric embeddings

Archetype = projection of the integral embedding on the cycle-space

Embedding = Choice of the basis B in Rp :

• Cycle vectors : given by the net voltages over the cycles of G

• Cut vectors : zero vectors Graphite layer : (63)

e1 1 0

A B A B e2

e3 0 -1

1 -1 0 1 0 K = 0 1 -1 B = 0 1 1 1 1 0 0

2/3 1/3 1/3 1 0 2/3 1/3 E = K-1.B = -1/3 1/3 1/3  0 1 = -1/3 1/3 -1/3 -2/3 1/3 0 0 -1/3 -2/3 Graphite layer : (63)

e2 b 2/3 1/3 e3 E = -1/3 1/3 -1/3 -2/3

e1 a

a.b = -1  (a, b) = 120◦ a.a = b.b = 2 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 A: 2a/3 + b/3 + T

B: x + e1 + 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 Topology of Crystal Structures

Advanced topics Nets as quotients of the minimal net

100

010 -1 -1 0 N (T = Z3) 001 N/<111>

<111> = translation subgroup generated by 111 N/<111>: T = <{100, 010}> Embeddings as projections of the archetype

A 3-d archetype, 2-d structures: 1-d kernel 1 4 2

2 3 C 3-cycle: 3.9 , 9 6 5 4-cycle: 4.82 D 3 B Nets as quotients (projections) A A Sr[Si ] 2

001 100 (I 41 32) 100 D 010 D 010 =^ 001 C B C B

A A

2 2 3 (4.8 ) (3.9 ,9 ) p3 m 1 01 p4 mm D 10 D 10 01 C B C B b

a c b

a c Nets as quotient graphs : kernel of the projection

• (N, T) a periodic net, • G = N/T, its quotient graph, • Voltage assignment  : N ~ G. • M[G] : minimal net, with translation group R. • w : closed walk of G with zero net voltage (w) = 0. • Lifting w in M[G] defines (w) in R • S = {(w) : (w) = 0}, subgroup of R S : Kernel of • N ~ M[G]/S the projection • S is generated by strong rings of N Rings and strong rings in graphs

• A ring is a cycle that is not the sum of two shorter cycles • A strong ring is a cycle that is not the sum of (an arbitrary number of) shorter cycles Example 1: the square net

A cycle A strong ring Sum of three strong rings Example 2: the cubic net

A strong ring: S1 A ring: R

A 6-cycle: C = S2 + S3

R = S1 + C

Rings are elements of the cycle-space: what is their origin, from the point of view of the labelled quotient graph? Rings of the derived graph

• Net voltage on a walk :

(w) = (e1) (e2 )...(en ) for w = e1e2...en • Cycles of G project on closed walks of G with net voltage 0 • No cycle in the covering tree! • Strong ring : cannot be written as a sum of smaller cycles Trivial rings

• G quotient graph with voltage assignment  in the free group Fn • Rings in M[G] are commutators of generators of Fn

[a,b]= aba-b- Nets as quotients: 1-periodic examples

M[B2] = square net

S =

N = M[B2]/S Square net :

F2, relator [a,b]

Quotient net :

2 -2 F2, relators [a,b], a b Check it! a b N

1 1 A 2-periodic case: tfa and tfa/<001>

100 010 10 01

001 Group presentation

Free group /C(r)

C(r) : consequence of the relation r, subgroup of , contains

r, its conjugates xr = xrx-1, their inverses and combinations Cycle-space : isomorphic to the kernel C(r)

Strong rings = shortest basis vectors of C(r) Cyclomatic number and generators

a, b: free generators Abelian group of rank 2 Commutation relationship: ab = ba

Commutator: r = [a, b] = aba-1b-1 [a, b2] = ab2a-1b-2 = (aba-1b-1)b(aba-1b-1 )b-1 = [a, b]b[a, b] a consequence of [a, b] Trivial rings : hcb

A faithfull representation

A single commutator [a, b]

A single strong ring, up to translation Fundamental rings : hex

Relator: abc (abc = 1 or c = b-1a-1 )

Two fundamental rings and no trivial ring:

abc acb 3-cycles

[a, b] , [a, c], [b, c] 4-cycles

acb → bac ( = b(acb)b-1 ) [a, b] = (abc)(bac)-1 Strong rings in kgm

Relators: ac-1 , bd-1 , [a, b]

[a, b] 8-cycle

[a, b] = aba-1b-1 = adc-1b-1 6-cycle Nets defined by a graph and relators, classification of nets

• G a graph with cyclomatic number n • Choose  : voltage assignment in A = Zn • S subgroup of A defined by a set of relators • N = M[G]/S • Nets N are ordered by set-inclusion of S subgroups in A. {2} Known nets derived from K3

Quartz adb-c- Sum of the 3 2-cycles NbO adb-c Sum of 2 3- cycles Moganite adc- Sum of 2 2- cycles afw bc- One 3-cycle Kagome ad , bc- Edge disjoint 3-cycles -W adb- , bc- Two tangent b 3-cycles {2} Lattice classification of nets derived from K3 Non-crystallographic nets

• Periodic nets (N,T) such that • Aut(N) is not isomorphic to any isometry group in the Euclidian space.

(Hidden symmetries = inconsistent with translations) Bounded automorphisms: B(N)

• g  Aut(N), such that: {d(x, g(x)) for x  V(N)} is bounded

• B(N) is normal in Aut(N). • For crystallographic nets: B = T (and conversely) Example t A0

B  = (A , B ) 0 0 0 -1t  T t (Xi) = Xi+1 for X = A, B Cu carboxylates in MOF structures



 : transposition of two Cu centers System of imprimitivity for B(N)

• Partition  of V(N) into finite subsets such that: • For any g  B(N) and v  (u), • g(v) and g(u) belong to the same cell (block) of 

A i blocks: C i {Ai, Bi} B i {Ci} Theorem

In any non-crystallographic net, the set F(N) of bounded automorphisms of finite order is a normal subgroup of the automorphism group: F(N) < Aut(N)  Orbits by F(N) provide a system of finite blocks of imprimitivity. F(N) and orbits

F(N) infinite, one fixed point-lattice

F(N) infinite, no fixed point-lattice

F(N) finite, no fixed point-lattice Equitable and equivoltage partitions

• A partition of the vertex set V(G) of a graph G with cells Ci

is equitable if the number of neighbours in Cj of a vertex u in

Ci is a constant bij, independent of u; • A vertex partition of a labelled quotient graph is equivoltage if: – it is equitable and – there is a 1-1 mapping between the stars of any two vertices of the same cell, which respects the voltages. Fundamental theorem of NC nets

Any non-crystallographic net can be represented by a labelled quotient graph with an equivoltage partition. Any periodic barycentric representation of the NC net displays vertex collisions; vertices that are equivalent under bounded automorphisms of finite order project on the same Euclidian point.  : Equivoltage partition

qT N N/T

q q q´ N/ T (N/)/T = (N/T)/

: system of imprimitivity derived from F(N) A 1-periodic example

1 qT 1 1

q q

q´T 1 Sequence of partitions

C E 1 1 1 q1 q2 1 A B A 1 1 B D 1 = {{A},{B},{C,D}} 2 = {{A},{B,E}} A 2-periodic case [E] 00 [D] 10 [C]

10 01 01 01 00

[I] 00 [A] 10 [B]

q

{A,C,E}

10 01

{B,D,F} (2) Creating a NC net from K3

A 10 10 01 B C 01

 = {{A},{B,C}} 4/4/o19 (Sowa, 2012)

ths Surgery Techniques

Elementary steps: vertex or edge addition and deletion

Combined operations:

Edge subdivision Vertex identification Vertex decoration Vertex decoration Decoration – contraction

Edge subdivision Vertex deletion

Decoration Vertex identification Contraction

Edge addition Graphs with crossings

Vertex identification K5 Decoration – Contraction in periodic nets Building units and underlying nets

Labelled quotient graph of building units: the voltage rank may be 0, 1, 2 or 3.

Labelled quotient graph of The underlying net: the voltage rank may be 0, 1, 2 or 3. Octahedra chains in rutile

rtl (rutile) labelled quotient graph sql – like interlinking of octahedra chains Centered sql From kgm to hca

Change of origin for A’ and C lattices From kgm to hca Relationships between hcb and hca ?

edge graph

? NiAs

Labelled quotient Labelled quotient graph of nia graph of MoS2 layers

Labelled quotient

graph of MgI2 layers Bi-layers

MoS2 MgI2 (kgd net) From hcb to MoS2 From hcb to kgd 001 hcb stacking in nia

equivalent vertices in MoS2 bi-layers 4 parallel hcb layers glued at

non-equivalent vertices in MgI2 bi-layers Garnet A3B2(XO4)3

Unconnected BO6 and XO4 units,

Two 3-periodic interpenetrated components for the A-O subnet.

Na3Sc2(VO4)3 Garnet (80 vertices – 192 edges)

Substitution of double A-O-A edges by single A-A edges with same net voltage lcv

T = A3O6 cages

Labelled quotient graph of one A-O component Decorated srs net Garnet: from srs to A-O subnets

Labelled quotient graph of lcv

T-decorated srs

Contraction (orange edges)

Change of origin

Contraction (orange edges) Garnet: local structure around A3O6 cages

Top and side views of 4-helicoidal chains srs-like interlinking of A3O6 cages o Tilt of A3O6 cages: arccos(√3/3) = 54.74

Q = 21°

B and X interlinking of the two Distortion of AO8 cubic coordination: Interpenetrated lcv A-O subnets  = p/2 - arccos(1/3) = 19.47o Groupoids

Translation in finite objects? (What is the symmetry of the chess board?)

Objects: S = [0,4] x [0,4], finite area

T = infinite translation group in the plane {G = {(x,t,y): x, y  S, t  T, x = ty} b (x,u,y)(y,v,z) = (x,uv,z) a Groupoids of action

G=(T,S) T==Z2 , S={A,B,C,D}

g=(C,b,B), h=(B,a,A) : gh=(C,ba,A)

Global symmetry Partial symmetry Local symmetry

m, t

m, g

gt t, g m Space Groupoid

A groupoid is formed from those symmetry operations (isometries) of a crystal structure built by isomorphic substrutures:

- Local Symmetries: symmetry operations of the substructure, - Partial Symmetries: symmetry operations mapping one substructure onto another, - Global Symmetries: symmetry operation of the entire structure. Structural analysis of pyroxenes

a = 9.5387 MgSiO3 b = 8.6601 HT clinoenstatite c = 5.2620 Space group: C2/c (Nº 15) b = 108.701

Mg1 Mg2+ 4 e 0. 0.90420(10) 0.25 Mg2 Mg2+ 4 e 0. 0.28560(10) 0.25 Si1 Si4+ 8 f 0.29223(6) 0.09198(7) 0.24570(10) O1 O2- 8 f 0.11300(10) 0.0828(2) 0.1380(3) O2 O2- 8 f 0.3636(2) 0.2577(2) 0.3179(3) O3 O2- 8 f 0.35430(10) 0.0065(2) 0.0258(3) Structural module in pyroxene (HT clinoenstatite, MgSiO3) in the plane (b-a,c) (primitive cell)

Simplified structure (Mg2 deleted) Stacking of modules in the direction a+b

a+b Interlinking of chains analysis through the labelled quotient graph

C(-110) B(-110)

B(-100) A C(010) A

C B

C kgd (kagome dual) HT clinoenstatite Quotient graph of the pyroxene module in pyroxenes (primitive cell)

-110 001 Layer group of the module

(a, b, c)  (b, a, c)

(b-a, c, a+b)  (a-b, c, a+b) (mx0 1/2 0) (x, y, z)  (-x, y+1/2, z)

(a, b, c)  (-b, -a, -c)

(b-a, c, a+b)  (b-a, -c, -a-b) 2x (x, y, z)  (x, -y, -z)

Crystal class 2/m (C2h) Layer group p2/b11 (monoclinic/rectangular) Nº 16 Layer group

-110

001 LT clinoenstatite: quotient graph and double modular structure in the plane (b, c)

Module HT clinoenstatite LT clinoenstatite versus HT clinoenstatite

E D Automorphism without fixed point:

 = (A, B)(C, F)(D, E)

Does not change voltages over cycles

A B  represents a translation of the periodic net mapping one module on the other

Distortion of the Crystal structure Along the respective direction C F Interlinking of chains and alternance of modules in LT clinoenstatite

C A E a u a u  b v c w b v c w

F B D  = 010; 2b = -110 New translation: Action of the automorphism  on the edges g = 010; 2 = 110  = ½ ½ 0 -11 -11 

kgd -10 01 Protoenstatite: quotient graph and double modular structure in the plane (b, c)

Module HT clinoenstatite Protoenstatite versus clinoenstatite

E D

 = (A, B)(C, F)(D, E)

Does not change voltages in the plane (a,b) but inverses the c axis.

A B

The linear part corresponds to a reflection Of the periodic net mapping one module on the other

C F Alternance of modules in protoenstatite

D B F

Proto Topology = kgd

E A C

Translation C A E Associated to : LT-clino ½ ½ 0 F B D Protoenstatite: inversion of chain in successive modules by n(½ ½ 0)[0,0,1]

c Ortoenstatite: quotient graph and quadruple modular structure in the plane (b, c)

E L G J

A D B C

F K H I Interlinking of chains and alternance of modules in ortoenstatite

c  a = (A,D,B,C)(E,L,G,J)(F,K,H,I) : automorphism of the reduced quotient graph Q, conserves u b b w a voltages along directions 100 e 010 v v w a u u c c b v Q u w w v c w a u b v New translation: b a  = 010; 4b = -120 c g = 010; 4 = 120 t = ¼ ½ 0 -11 -11 

kgd -10 01 Sequence of modules in ortoenstatite

 does not represent an automorphism of the net:

 inverses the c axis, and infinite linear chains, alternatively, two modules yes, two modules no. b t

 represents a partial symmetry operation in a groupoid.

a Space groupoid of ortoenstatite

• Translation t = t(¼ ½ 0) • Glide reflection  = (¼ ½ 0)[0,0,1]

• Modules : mk, k  Z (k-k/2) k/2 • mk = t  .m0 = k.m0 • k/2: largest integer inferior or equal to k/2 -1 • Layer group of module mk: k.p2/b11.k -1 • Partial symmetry operations from mk to ml: l.p2/b11.k

• Only 2k act upon the complete structure: k 2k = (t) → glide reflection a → Pbca Thank you for your attention Main references

• Beukemann A., Klee W.E., Z. Krist. 201, 37-51 (1992) • Blatov V.A., Acta. Cryst. A56, 178-188 (2000) • Blatov, V.A., Shevchenko, A.P., Proserpio, D.M., Cryst. Growth Des. 14, 3576- 3586 (2014) • Carlucci L., Ciani G., Proserpio D.M., Coord. Chem. Rev. 246, 247-289 (2003) • Chung S.J., Hahn Th., Klee WE., 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) • 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 A61, 501-511 (2005) • Eon, J.-G., Acta Cryst A67, 68—86 (2011) • Eon, J.-G., Acta Cryst A72, 268-293 (2016) • Gross J.L, Tucker T.W., Topological Graph Theory, Dover (2001) • Harary F., Graph Theory, Addison-Wesley (1972) • Klee WE., Cryst. Res. Technol., 39(11), 959-960 (2004)