Semisymmetry of Generalized Folkman Graphs

A regular edge but not vertex transitive graph is said to be semisym metric The study of semisymmetric graphs was initiated by Folkman who among others gave constructions of several in nite families such graphs In this paper a generalization of his construction for or ders a multiple of is proposed giving rise to some new families of semisymmetric graphs In particular one associated with the cyclic group of order n n which belongs to the class of tetracirculants that is graphs admitting an automorphism with precisely four orbits all of the same length Semisymmetry properties of tetracirculants are investigated in greater detail leading to a classi cation of all semisym metric graphs of order p p a prime Introductory remarks Throughout this paper graphs are assumed to be nite simple and unless speci ed otherwise undirected By p we shall always denote an odd prime For the group theoretic concepts and notation not de ned here we refer the reader to Given a graph X we let V X E X and AutX be the vertex set the edge set and the automorphism group of X respectively We say that X is vertex transitive and edge transitive if AutX acts transitively on V X and E X respectively It is easily seen that an edge but not vertex transitive graph X is necessarily bipartite where the two parts of the bipartition are orbits of AutX Moreover if X is regular then these two parts have equal cardinality A regular edge but not vertex transitive graph is called semisymmetric The study of semisymmetric graphs was initiated by Folkman who gave a construction of several in nite families of such graphs including among others a family of semisymmetric graphs of order p p an odd prime The smallest graph in this construction has vertices and happens to be the smallest semisymmetric graph Insipired by Folkman s work the study of semisymmetric graphs has recently received a wide attention resulting in a number of published articles see One of the main purposes of this article is to generalize Folkman s constructions We start by introducing the concept of orbital digraphs Let H be a transitive permutation group acting on a set V and let v V There is a correspondence between the set of suborbits of H that is the set of orbits of the stabilizer Hv on V and the set of orbitals of H that is the set of orbits in the natural action of H on V V with the trivial suborbit fvg corresponding to the diagonal f v v v V g For an orbital we let S v fw j v w g denote the suborbit of H relative to v associated with Conversely for a suborbit S of H relative to v we let S v be the associated orbital in the above correspondence The paired orbital of an orbital is the orbital f v w w v g If we say that is a self paired orbital Similarly for a suborbit S of H relative to v we let S S v denote the paired suborbit of S If S S we say that S is self paired The orbital digraph X H V of H V relative to is the digraph with vertex set V and arc set The underlying undirected graph of X H V will be called the orbital graph of H V relative to and will be denoted by X H V If is a self paired orbital then X H V admits a vertex and arc transitive action of H On the other hand if is not self paired then X H V admits a vertex and edge but not arc transitive action of H in short a arc transitive action of H For a permutation of V contained in the normalizer of the permutation group H in the symmetric group SymV we let denote the set f x y j x y g Since normalizesH the set is also an orbital of H If v V is left xed by and S S v then the set S fs j s Sg is the suborbit S v of H which corresponds to the orbital relative to the vertex v The following construction starting with a transitive permutation group H and its orbital is a generalization of Folkman s construction of semisym metric graphs arising from abelian groups see Theorem De nition Let H be a transitive permutation group on a set V let be its orbital let k be an integer and let be a permutation of V contained in the normalizer ofH in SymV and such that k H Let B fBx j x V g and V j fx j j x V g j Z k be k copies of the set V Let Y H V k denote the graph with vertex set B V V k and edge set fx jBy j j Z k x y j g Furthermore let V j fx j j x V g j Z k be k copies of the set V The generalized Folkman graph F H V k has vertex set S i j Z k Vij and edge set fx jy i j i j Z k x y j g Observe that the generalized Folkman graph F H V k is obtained from Y H V k by expanding each Bx to a k tuple of vertices x x x k each retaining the neighbors set of Bx For the generalized Folkman graph F H V k it will be sometimes convenient to specify a suborbit S corresponding to rather than itself The notation F H V S k will then be used instead of F H V k The same applies for the graph Y H V k Note that the generalized Folkman graphs are all regular and bipartite Furtheremore letting G hH i we can see that every element of G induces an automorphism of F H V k withH stabilizing all the sets Vij i Z j Z k and stabilizing the sets V j and cyclically permuting the sets V j j Z k With abuse of notation the symbols H and G will also denote the corresponding induced actions on F H V k and Y H V k Observe that for every x V the vertices x j j Z k have the same neighbors sets in the graph F F H V k It follows that for each x V the automorphism group AutF contains a copy of the symmetric group Sk xing the sets V j j Z k pointwise and acting on the set fx j j j Z kg by permuting the indices j Z k The group generated by G and these automorphisms acts transitively on the set of edges of F and has two orbits in its action on the set of vertices of F namely S j Z k V j and S j Z k V j The generalized Folkman graph F is therefore a regular bipartite edge transitive graph with at most two vertex orbits The fact that for each vertex in S j Z k V j there are at least k other vertices in S j Z k V j sharing the same set of neighbors in F gives rise to the following simple su cient condition for the semisymmetry of generalized Folkman graphs The proof is straightforward and is omitted Proposition If no k distinct vertices in S j Z k V j have the same set of neighbors in the graph Y H V k then the generalized Folkman graph F H V k is semisymmetric One of the main goals of this article is to give constructions of several in nite families of semisymmetric generalized Folkman graphs see Section Some of these constructions are immediate generalizations of the original Folkman s constructions of semisymmetric graphs of valency correspond ing to abelian groups Examples and The others are new and arise in the context of alternating groups Examples and We remark that the generalized Folkman graphs of the last two examples are associated with certain graphs admitting arc transitive group actions Namely let X be a graph admitting a arc transitive action of a subgroup H of AutX and an arc transitive action of a subgroup G of AutX where H is of in dex in G Then there exists a non self paired orbital of H such that X X H V X and is an orbital of G Now let be an ar bitrary element in G n H Then of course H and we can construct the generalized Folkman graph F H V X Conversely the generalized Folkman graph F H V gives rise to a graph admit ting a arc transitive action of H and an arc transitive action of G hH i provided is not self paired and is the paired orbital of The above described connection between these two families of graphs is the con tent of the sequel to this paper In Section we give a number of preliminary results on n polycirculants that is graphs addmiting an automorphism having precisely n orbits all of equal size These results are used in Section which is devoted to the study of semisymmetry properties of polycirculants Finally building on the results from these two sections we classify and solve the isomorphism problem for semisymmetric graphs of order p p a prime in Section In particular we show that every such graph is a generalized Folkman graph