# On even-closedness of vertex-transitive graphs

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1 On even-closedness of vertex-transitive graphs Akademgorodok, August 2016

2 Overview Vertex-transitive graphs from pairs of groups Determining the full automorphism group Symmetry through odd automorphisms (STOA): Core research problem Cubic symmetric graphs

3 Vertex-transitive graphs An automorphism of a graph X = (V, E) is an isomorphism of X with itself. Thus each automorphism α of X is a permutation of the vertex set V which preserves adjacency. A graph X is vertex-transitive if its automorphism group Aut(X ) acts transitively on vertices.

4 Vertex-transitive graphs - alternative (constructive) definition Let (G, H) be a pair of abstract groups such that H G, G/H... the set of left cosets of G with respect to H. G acts on G/H by left multiplication as a transitive permutation group. This action induces an action of G on G/H G/H; the corresponding orbits are called orbitals. Define a graph with vertex set G/H and the edge set E = Clo(O), where O is a union of orbitals and Clo(O) a symmetric closure of O, that is, if (x, y) O then both (x, y) and (y, x) are in Clo(O). This graph is vertex-transitive, and every vertex-transitive graph can be obtained in this way.

5 Example - the cube Q 3 G = S 4 H = (1 2 3) = Z 3 V = S 4 /Z 3 E = O, where O is the orbital containing the pair (H, (1 4)H). (14)H (142)H H (1234)H (14)(23)H (12) H (1324)H (143)H

6 The cube Q 3 - Alternatively The cube can also be obtained from pair G = Z 3 2 and H = 1.

7 The cube Q 3 - Alternatively The cube can also be obtained from pair G = Z 3 2 and H = 1. In general, we say that a vertex-transitive graph X = Cay(G, S) is a Cayley graph if it can be obtained from a pair (G, 1) where S = S 1 G \ {1} denotes the set of neighbors of 1 in X. Q 3 = Cay(Z 3 2, {100, 010, 001})

8 The full automorphism group of a vertex-transitive graph If X is a vertex-transitive graph arising from a pair (G, H) then G Aut(X ). What is Aut(X ), i.e. when is G = Aut(X )?

9 The full automorphism group of a vertex-transitive graph If X is a vertex-transitive graph arising from a pair (G, H) then G Aut(X ). What is Aut(X ), i.e. when is G = Aut(X )? Back to the cube Q 3 : Clearly, Z 3 2 Aut(Q 3) and S 4 Aut(Q 3 ). Is G = S 4 the full automorphism group of the cube?

10 The full automorphism group of a vertex-transitive graph If X is a vertex-transitive graph arising from a pair (G, H) then G Aut(X ). What is Aut(X ), i.e. when is G = Aut(X )? Back to the cube Q 3 : Clearly, Z 3 2 Aut(Q 3) and S 4 Aut(Q 3 ). Is G = S 4 the full automorphism group of the cube? Aut(Q 3 ) = S 4 Z 2.

11 ECOVTG Even-closedness of vertex-transitive graphs A crucial question in algebraic graph theory and beyond: Given a graph, are there any symmetries beyond the obvious ones, and, if yes, how can one determine the full set? We approach this question by building on the duality of even/odd permutations associated with graphs.

12 Even/odd automorphisms An automorphism of a graph is said to be even/odd if it acts on the set of vertices as an even/odd permutation.

13 The Petersen graph

14 The Petersen graph Obvious automorphisms: rotation reflection Semi-obvious automorphism: swap (inner/outer 5-cycle) There exist additional automorphisms (mixers), disrespecting inner and outer cycles.

15 The Petersen graph Aut(X ) = S 5

16 ECOVTG Even-closedness of vertex-transitive graphs The full automorphism group of the Petersen graph contains involutions with three orbits of size 2 and four fixed vertices, and hence odd (as permutations) automorphisms.

17 The cube Q 3 All automorphisms of the cube are even.

18 ECOVTG Even-closedness of vertex-transitive graphs 2-closure H (2) of H = intersection of automorphism groups of all basic orbital graphs of H (basic = arising from single orbitals) H is 2-closed if H (2) = H. Even group = group with only even permutations. Odd group = group containing also odd permutations. Question For H even, can we imbed it into an odd group via basic orbital graphs? If H is not 2-closed, is the 2-closure H (2) odd? If H (2) even, is there at least one basic orbital graph of H admitting an odd automorphism? If yes H is orbital-odd, otherwise H is even-closed.

19 ECOVTG Even-closedness of vertex-transitive graphs Back to the cube and the Petersen graph: The Petersen graph: For H = A 5 the 2-closure is S 5, and is odd. So H = A 5 as a group of degree 10 is orbital-odd. The cube: For H {Z 3 2, S 4} the 2-closure is S 4 Z 2, and is even. Still, H {Z 3 2, S 4} as a group of degree 8 is orbital-odd. F102 a cubic symmetric graph of type {4 1 }: Its automorphism group is isomorphic to PSL(2, 17). Magma calculations show that PSL(2, 17) as a group of degree 102 is even-closed..

20 Orbital-odd degrees / VTG with odd automorphisms For some integers n, all transitive groups of degree n are orbital-odd. For example, 2p, where p is a prime, is such an integer.

21 Odd automorphisms in VT graphs of order 2p, p prime X VTG of order 2p, G AutX transitive G is imprimitive, blocks of size p; G is imprimitive, blocks of size 2, G is primitive.

22 Odd automorphisms in VT graphs of order 2p, p prime X VTG of order 2p, G AutX transitive

23 Odd automorphisms in VT graphs of order 2p, p prime X VTG of order 2p, G AutX transitive G is imprimitive, blocks of size p a swap exists as an immediate consequence of existence of two blocks,

24 Odd automorphisms in VT graphs of order 2p, p prime X VTG of order 2p, G AutX transitive G is imprimitive, blocks of size p a swap exists as an immediate consequence of existence of two blocks, G is imprimitive, blocks of size 2 a swap exists as a consequence of existence of a smaller H G with blocks of size p,

25 Odd automorphisms in VT graphs of order 2p, p prime X VTG of order 2p, G AutX transitive G is imprimitive, blocks of size p a swap exists as an immediate consequence of existence of two blocks, G is imprimitive, blocks of size 2 a swap exists as a consequence of existence of a smaller H G with blocks of size p, G is primitive a swap exists as a consequence of the classification of finite simple groups (CFSG).

26 Odd automorphisms in VT graphs of order 2p, p prime X VTG of order 2p, G AutX transitive G is imprimitive, blocks of size p a swap exists as an immediate consequence of existence of two blocks, G is imprimitive, blocks of size 2 a swap exists as a consequence of existence of a smaller H G with blocks of size p, G is primitive a swap exists as a consequence of the classification of finite simple groups (CFSG). All swaps are odd automorphisms!

27 G is primitive Example: p = 5, A 5, S 5 acting on pairs from {1, 2, 3, 4, 5}. Associated graphs: the Petersen graph and its complement. By CFSG a primitive group of degree 2p, p > 5, is double transitive, and as such the corresponding orbital graph is the complete graph K 2p. No CFSG-free proof of this fact exists. Also, no CFSG-free answer to the even/odd question for VTG of order 2p exists.

28 Cay(D 16, {ρ ±1, τ}) and Cay(Z 16, {±1, 2k}) Two examples of cubic vertex-transitive graphs, one with and one without odd automorphisms.

29 Cayley graphs I Proposition A Cayley graph on a group G admits odd automorphisms in the left regular representation G L of G if and only if Sylow 2-subgroups of G are cyclic. In particular, a Cayley graph of order 2 (mod 4) admits odd automorphisms.

30 Cayley graphs II Proposition Let X = Cay(G, S) be a Cayley graph on an abelian group G and let τ Aut(G) be such that τ(i) = i. Then G L, τ Aut(X ), and there exists an odd automorphism in G L, τ if and only if one of the following holds: G 3 (mod 4), G 2 (mod 4), G 0 (mod 4) and a Sylow 2-subgroup of G is cyclic.

31 Cubic symmetric graphs A graph X is called symmetric if its automorphism group acts transitively on its vertex set as well as its arc set. arc = directed edge

32 Cubic symmetric graphs A symmetric graph X is said to be s-regular if for any two s-arcs in X, there is a unique automorphism of X mapping one to the other. Tutte, 1947 Every finite cubic symmetric graph is s-regular for some s 5. The list of all possible pairs of vertex and edge stabilizers in cubic s-regular graphs: s Aut(X ) v Aut(X ) e 1 Z 3 id 2 S 3 Z 2 2 or Z 4 3 S 3 Z 2 D 8 4 S 4 D 16 or QD 16 5 S 4 Z 2 (D 8 Z 2 ) Z 2 The vertex stabilizer is of order 3 2 s 1 in a cubic s-regular graph.

33 17 types of cubic symmetric graphs (Conder, Nedela, 2009) s Type Bipartite? s Type Bipartite? s Type Bipartite? 1 {1} Sometimes 3 {2 1, 3} Never 5 {1, 4 1, 4 2, 5} Always 2 {1, 2 1 } Sometimes 3 {2 2, 3} Never 5 {4 1, 4 2, 5} Always 2 {2 1 } Sometimes 3 {3} Sometimes 5 {4 1, 5} Never 2 {2 2 } Sometimes 4 {1, 4 1 } Always 5 {4 2, 5} Never 3 {1, 2 1, 2 2, 3} Always 4 {4 1 } Sometimes 5 {5} Sometimes 3 {2 1, 2 2, 3} Always 4 {4 2 } Sometimes Examples: K 4 = {1, 2 1 }, K 3,3 = {1, 2 1, 2 2, 3}, Q 3 = {1, 2 1 }, F 010A = {2 1, 3}, F 014A = {1, 4 1 }, F 016A = {1, 2 1 }, F 018A = {1, 2 1, 2 2, 3}, F 020A = {1, 2 1 }, F 020B = {2 1, 2 2, 3}.

34 The four smallest cubic symmetric graphs Odd automorphisms? YES YES NO YES

35 The Heawood graph F 014A of type {1, 4 1 } Odd automorphism? YES

36 The Möbius-Kantor graph F 016A of type {1, 2 1 } Odd automorphism? NO

37 The Pappus graph F 018A of type {1, 2 1, 2 2, 3} Odd automorphism? YES

38 Automorphism group of a cubic symmetric graph The automorphism group of any finite cubic symmetric graph is an epimorphic image of one of the following seven groups: G 1 = h, a h 3 = a 2 = 1, G 1 2 = h, a, p h 3 = a 2 = p 2 = 1, apa = p, php = h 1, G 2 2 = h, a, p h 3 = p 2 = 1, a 2 = p, php = h 1, G 3 = h, a, p, q h 3 = a 2 = p 2 = q 2 = 1, apa = q, qp = pq, ph = hp, php = h 1, G 1 4 = h, a, p, q, r h 3 = a 2 = p 2 = q 2 = r 2 = 1, apa = p, aqa = r, h 1 ph = q, h 1 qh = pq, rhr = h 1, pq = qp, pr = rp, rq = pqr, G 2 4 = h, a, p, q, r h 3 = p 2 = q 2 = r 2 = 1, a 2 = p, a 1 qa = r, h 1 ph = q, h 1 qh = pq, rhr = h 1, pq = qp, pr = rp, rq = pqr, G 5 = h, a, p, q, r, s h 3 = a 2 = p 2 = q 2 = r 2 = s 2 = 1, apa = q, ara = s, h 1 ph = p, h 1 qh = r, h 1 rh = pqr, shs = h 1, pq = qp, pr = rp, ps = sp, qr = rq, qs = sq, sr = pqrs.

39 Odd automorphisms in cubic symmetric graph Theorem: Let X be a cubic symmetric graph of order 2n. Then Table below gives a full information on existence of odd automorphisms in X. Type Odd automorphisms exist if and only if {1} n odd {1, 2 1 } n odd, or n = 2 k 1 (2t + 1) and X is a (2t+1)-Cayley graph on a cyclic group of order 2 k, where k 2 {2 1 } n odd and X bipartite {2 2 } never {1, 2 1, 2 2, 3} n odd {2 1, 2 2, 3} n odd {2 1, 3} n odd {2 2, 3} n odd {3} n odd and X bipartite {1, 4 1 } n odd {4 1 } n odd and X bipartite {4 2 } n odd {1, 4 1, 4 2, 5} n odd {4 1, 4 2, 5} n odd {4 1, 5} never {4 2, 5} never {5} n odd and X bipartite

40

41 THANK YOU! HVALA!

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