# Graphs with a high degree of symmetry

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1 Graphs with a high degree of symmetry Robert Gray University of Leeds June 14th 2007

2 Outline Introduction Graphs, automorphisms, and vertex-transitivity Two notions of symmetry Distance-transitive graphs Homogeneous graphs An intermediate notion Connected-homogeneous graphs

3 Outline Introduction Graphs, automorphisms, and vertex-transitivity Two notions of symmetry Distance-transitive graphs Homogeneous graphs An intermediate notion Connected-homogeneous graphs

4 Graphs and automorphisms Definition A graph Γ is a pair (VΓ, EΓ) VΓ - vertex set EΓ - set of 2-element subsets of VΓ, the edge set. If {u, v} EΓ we say that u and v are adjacent writing u v. The neighbourhood of u is Γ(u) = {v VΓ : v u}, and the degree (or valency) of u is Γ(u). A graph Γ is finite if VΓ is finite, and is locally-finite if all of its vertices have finite degree. An automorphism of Γ is a bijection α : VΓ VΓ sending edges to edges and non-edges to non-edges. We write G = Aut Γ for the full automorphism group of Γ.

5 Graphs with symmetry Roughly speaking, the more symmetry a graph has the larger its automorphism group will be (and vice versa). Aim. To obtain classifications of families of graphs with a high degree of symmetry. In each case we impose a symmetry condition P and then attempt to describe all (countable) graphs with property P. For each class, this naturally divides into three cases: finite graphs; infinite locally-finite graphs; infinite non-locally-finite graphs.

6 Vertex-transitive graphs Definition Γ is vertex transitive if G acts transitively on VΓ. That is, for all u, v VΓ there is an automorphism α G such that u α = v. This is the weakest possible condition and there are many examples. Complete graph K r has r vertices and every pair of vertices is joined by an edge. Cycle C r has vertex set {1,..., r} and edge set {{1, 2}, {2, 3},..., {r, 1}}. Empty graph I r is the complement of the complete graph K r. (The complement Γ of Γ is defined by VΓ = VΓ, EΓ = {{i, j} : {i, j} EΓ}). K 4 C 6

7 Some vertex transitive bipartite graphs Definition A graph is called bipartite if the vertex set may be partitioned into two disjoint sets X and Y such that no two vertices in X are adjacent, and no two vertices of Y are adjacent. Complete bipartite every vertex in X is adjacent to every vertex of Y (written K a,b if X = a, Y = b). Perfect matching there is a bijection π : X Y such that EΓ = {{x, π(x)} : x X} Complement of perfect matching {x, y} EΓ y π(x) x π (x) K 3,3

8 Cayley graphs of groups Definition G - group, A G a generating set for G such that 1 G A and A is closed under taking inverses (so x A x 1 A). The (right) Cayley graph Γ = Γ(G, A) is given by VΓ = G; EΓ = {{g, h} : g 1 h A}. Thus two vertices are adjacent if they differ in G by right multiplication by a generator. Fact. The Cayley graph of a group is always vertex transitive.

9 Cayley graph Example (Cayley graph of S 3 ) G = the symmetric group S 3, A = {( 1 2 ), ( 2 3 ), ( 1 3 )} ( ) ( ) ( ) ( 1 2 ) ( 2 3 ) ( 1 3 ) Γ(G, A) = K 3,3 a complete bipartite graph.

10 Vertex-transitive graphs On the other hand, not every vertex transitive graph arises in this way. Example (Petersen graph) The Petersen graph is vertex transitive but is not a Cayley graph. There are far too many vertex transitive graphs for us to stand a chance of achieving a classification.

11 Outline Introduction Graphs, automorphisms, and vertex-transitivity Two notions of symmetry Distance-transitive graphs Homogeneous graphs An intermediate notion Connected-homogeneous graphs

12 Distance-transitive graphs Definition In a connected graph Γ we define the distance d(u, v) between u and v to be the length of a shortest path from u to v. Definition A graph is distance-transitive if for any two pairs (u, v) and (u, v ) with d(u, v) = d(u, v ), there is an automorphism taking u to u and v to v. distance-transitive vertex-transitive Example A connected finite distance-transitive graph of valency 2 is simply a cycle C n.

13 Hamming graphs and hypercubes A family of distance-transitive graphs Definition The Hamming graph H(d, n). Let Z n = {0, 1, 2,..., n 1}. Then the vertex set of H(d, n) is Z d n = Z n Z n }{{} d times and two vertices u and v are adjacent if and only if they differ in exactly one coordinate. The d-dimensional hypercube is defined to be Q d := H(d, 2). Its vertices are d-dimensional vectors over Z 2 = {0, 1}. Fact. H(d, n) is distance transitive

14 Hypercubes Q i (i = 2, 3, 4) (1,0) (1,1) (1,1,1,1) (0,0) Q 2 (0,1) (1,1,0) (1,1,1) (1,0,0) (0,1,0) (1,0,1) (0,1,1) (0,0,0) (0,0,1) (0,0,0,0) Q 4 Q 3

15 Finite distance-transitive graphs The classification of the finite distance-transitive graphs is still incomplete, but a lot of progress has been made. Definition A graph is imprimitive if there is an equivalence relation on its vertex set which is preserved by all automorphisms.

16 Imprimitive distance-transitive graphs The cube is imprimitive in two different ways. 1. Bipartite The bipartition relation u v d(u, v) is even is preserved (2 equivalence classes: red and blue) 2. Antipodal The relation u v u = v or d(u, v) = 3 is preserved (4 equivalence classes: black, blue, purple and red)

17 Smith s reduction Smith (1971) showed that the only way in which a finite distance-transitive graph (of valency > 2) can be imprimitive is as a result of being bipartite or antipodal (as in the cube example above). This reduces the classification of finite distance-transitive graphs to: 1. classify the finite primitive distance-transitive graphs (this is close to being complete, using the classification of finite simple groups; see recent survey by John van Bon in European J. Combin.); 2. find all bipartite doubles and antipodal covers of these graphs (still far from complete).

18 Infinite locally-finite distance-transitive graphs Trees Definition (Tree) A tree is a connected graph without cycles. A tree is regular if all vertices have the same degree. We use T r to denote a regular tree of valency r. Fact. A regular tree T r (r N) is an example of an infinite locally-finite distance-transitive graph. Definition (Semiregular tree) T a,b : A tree T = X Y where X Y is a bipartition, all vertices in X have degree a, and all in Y have degree b. A semiregular tree will not in general be distance transitive.

19 Locally finite infinite distance-transitive graphs A family of examples!! \$ \$ F F V V % % D D H H T T ' ' ) ) / / Let r 1 and l 2 be integers. Take a bipartite semiregular tree T r+1,l one block A with vertices of degree r + 1 the other B with vertices of degree l Define X r,l Vertex set = B b 1, b 2 B joined iff they are at distance 2 in T r+1,l. L L N N P P R R T 3,4

20 Locally finite infinite distance-transitive graphs A family of examples!! \$ \$ F F V V % % D D H H T T ' ' ) ) / / Example X r,l = X 2,4. Let r = 2 and l = 4. So T r+1,l = T 3,4 A = vertices of degree 3 (in black) B = vertices of degree 4 (in red) X 2,4 Vertex set = B = red vertices b 1, b 2 B joined iff they are at distance 2 in T 3,4. L L N N P P R R T 3,4

21 Locally finite infinite distance-transitive graphs A family of examples Example X r,l = X 2,4. Let r = 2 and l = 4. So T r+1,l = T 3,4 A = vertices of degree 3 (in black) B = vertices of degree 4 (in red) X 2,4 Vertex set = B = red vertices b 1, b 2 B joined iff they are at distance 2 in T 3,4.

22 Locally finite infinite distance-transitive graphs A family of examples Example X r,l = X 2,4. Let r = 2 and l = 4. So T r+1,l = T 3,4 A = vertices of degree 3 (in black) B = vertices of degree 4 (in red) X 2,4 Vertex set = B = red vertices b 1, b 2 B joined iff they are at distance 2 in T 3, X 2,4

23 Macpherson s theorem The graphs X κ1,κ 2 (κ 1, κ 2 (N \ {0}) {ℵ 0 }) are distance transitive. The neighbourhood of a vertex consists of κ 2 copies of the complete graph K κ1. Theorem (Macpherson (1982)) A locally-finite infinite graph is distance transitive if and only if it is isomorphic to X k,r for some k, r N. The key steps in Macpherson s proof are to take an infinite locally finite distance-transitive graph Γ and 1. prove that Γ is tree-like (i.e. show Γ has infinitely many ends) 2. apply a powerful theorem of Dunwoody (1982) about graphs with more than one end

24 Non-locally-finite infinite distance-transitive graphs On the other hand, for infinite non-locally-finite distance-transitive graphs far less is known. The following result is due to Evans. Theorem There exist 2 ℵ 0 non-isomorphic countable distance-transitive graphs. Proof. Makes use of a construction originally due to Hrushovski (which is itself a powerful strengthening of Fraïssé s method for constructing countable structures by amalgamation).

25 Outline Introduction Graphs, automorphisms, and vertex-transitivity Two notions of symmetry Distance-transitive graphs Homogeneous graphs An intermediate notion Connected-homogeneous graphs

26 Homogeneous graphs Definition A graph Γ is called homogeneous if any isomorphism between finite induced subgraphs extends to an automorphism of the graph. Homogeneity is the strongest possible symmetry condition we can impose. homogeneous distance-transitive vertex-transitive

27 A finite homogeneous graph Definition (Line graph) The line graph L(Γ) of a graph Γ has vertex set the edge set of Γ, and two vertices e 1 and e 2 joined in L(Γ) iff the edges e 1, e 2 share a common vertex in Γ. Example L(K 3,3 ) is a finite homogeneous graph a b c d a b c d e f e f

28 Classification of finite homogeneous graphs Gardiner classified the finite homogeneous graphs. Theorem (Gardiner (1976)) A finite graph is homogeneous if and only if it is isomorphic to one of the following: 1. finitely many disjoint copies of K r (r 1) (or its complement); 2. The pentagon C 5 ; 3. Line graph L(K 3,3 ) of the complete bipartite graph K 3,3.

29 Infinite homogeneous graphs Definition (The random graph R) Constructed by Rado in The vertex set is the natural numbers (including zero). For i, j N, i < j, then i and j are joined if and only if the ith digit in j (in base 2) is 1. Example Since 88 = = the numbers less that 88 that are adjacent to 88 are just {3, 4, 6}. Of course, many numbers greater than 88 will also be adjacent to 88 (for example 2 88 will be).

30 The random graph Consider the following property of graphs: (*) For any two finite disjoint sets U and V of vertices, there exists a vertex w adjacent to every vertex in U and to no vertex in V. Theorem There exists a countably infinite graph R satisfying property (*), and it is unique up to isomorphism. The graph R is homogeneous. Existence. The graph R defined above satisfies property (*). Uniqueness and homogeneity. Both follow from a back-and-forth argument. Property (*) is used to extend the domain (or range) of any isomorphism between finite substructures one vertex at a time.

31 Fraïssé s theorem Definition A relational structure M is homogeneous if any isomorphism between finite induced substructures of M extends to an automorphism of M. The age of M is the class of isomorphism types of its finite substructures. Fraïssé (1953) showed how to recognise the existence of homogeneous structures from their ages. A class C is the age of a countable homogeneous structure M if and only if C is closed under isomorphism, closed under taking substructures, contains only countably many structures up to isomorphism, and satisfies the amalgamation property. If these conditions hold, then M is unique, C is called a Fraïssé class, and M is called the Fraïssé limit of the class C.

32 Picture of amalgamation The amalgamation property says that two structures in C with isomorphic substructures can be glued together, inside a larger structure of C, in such a way that the substructures are identified. B 2 B 2 C A B A A B 1 (AP) Given B 1, B 2 C and isomorphism f : A 1 A 2 with A i B i (i = 1, 2), C C in which B 1 and B 2 are embedded so that A 1 and A 2 are identified according to f.

33 Countable homogeneous graphs Examples The class of all finite graphs is a Fraïssé class. Its Fraïssé limit is the random graph R. The class of all finite graphs not embedding K n (for some fixed n) is a Fraïssé class. We call the Fraïssé limit the countable generic K n -free graph. Theorem (Lachlan and Woodrow (1980)) Let Γ be a countably infinite homogeneous graph. Then Γ is isomorphic to one of: the random graph, a disjoint union of complete graphs (or its complement), the generic K n -free graph (or its complement).

34 Outline Introduction Graphs, automorphisms, and vertex-transitivity Two notions of symmetry Distance-transitive graphs Homogeneous graphs An intermediate notion Connected-homogeneous graphs

35 Connected-homogeneous graphs Distance-transitive graphs - classification incomplete Homogeneous graph - classified Question. Do there exist natural classes between homogeneous and distance-transitive that can be classified? Definition A graph Γ is connected-homogeneous if any isomorphism between connected finite induced subgraphs extends to an automorphism. homogeneous connected-homogeneous distance-transitive

36 Finite connected-homogeneous graphs Gardiner classified the finite connected-homogeneous graphs. Theorem (Gardiner (1978)) A finite graph is connected-homogeneous if and only if it is isomorphic to a disjoint union of copies of one of the following: 1. a finite homogeneous graph 2. complement of a perfect matching 3. cycle C n (n 5) 4. the line graph L(K s,s ) of a complete bipartite graph K s,s (s 3) 5. Petersen s graph 6. the graph obtained by identifying antipodal vertices of the 5-dimensional cube Q 5

37 Infinite connected-homogeneous graphs Theorem (RG, Macpherson (in preparation)) Any countable connected-homogeneous graph is isomorphic to the disjoint union of a finite or countable number of copies of one of the following: 1. a finite connected-homogeneous graph; 2. a homogeneous graph; 3. the random bipartite graph; 4. the complement of a perfect matching; 5. the line graph of a complete bipartite graph K ℵ0,ℵ 0 ; 6. a graph X κ1,κ 2 with κ 1, κ 2 (N \ {0}) {ℵ 0 }. (The proof relies on the Lachlan-Woodrow classification of fully homogeneous graphs.)

38 Possible future work Consider connected-homogeneity for other kinds of relational structure. Schmerl (1979) classified the countable homogeneous posets. It turns out that weakening homogeneity to connected-homogeneity here essentially gives rise to no new examples. Theorem (RG, Macpherson (in preparation)) A countable poset is connected-homogeneous if and only if it is isomorphic to a disjoint union of a countable number of isomorphic copies of some homogeneous countable poset. The corresponding result for digraphs seems to be difficult.

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