Let S be a subset of the units in n . Let # be a circulant graph of order n (a Cayley graph of n ) such that if ij E(#), then i j (mod n) S.Toida conjectured that if # # is another circulant graph of order n,then#and# # are isomorphic if and only if they are isomorphic by a group automorphism of n .In this paper, we prove that Toida's conjecture is true. We further prove that Toida's conjecture implies Zibin's conjecture, a generalization of Toida's conjecture.

Let ϕ be Euler’s phi function. We prove that a vertex-transitive graph Ɣ of order n, with gcd(n, ϕ(n)) = 1, is isomorphic to a circulant graph of order n if and only if Aut(Ɣ) contains a transitive solvable subgroup. As a corollary, we prove that every vertex-transitive graph Ɣ of order n is isomorphic to a circulant graph of order n if and only if for every such Ɣ, Aut(Ɣ) contains a transitive solvable subgroup and n = 4, 6, or gcd(n, ϕ(n)) = 1. c ○ 2000 Academic Press In recent years, a large amount of interest has been shown in vertex-transitive graphs, that is, graphs whose automorphism group acts transitively on the vertices of the graph. Much of this interest has been aimed, in one form or another, at the problem of classifying vertex-transitive graphs in terms of ‘small ’ subgroups of their automorphism groups (see, for example [2, 7, 8, 11–13] for some such results). This problem, in large part, has been made accessible by the recent classification of the finite simple groups, which allows one to find for many values of n, all transitive groups of degree n that are primitive, but not doubly transitive. One may thus determine those graphs whose automorphism group acts primitively on the vertex set. In this paper we will take, in some sense, the opposite approach. Namely, we examine vertextransitive

Abstract. We explicitly determine all of the transitive groups of degree p 2, p a prime, whose Sylow p-subgroup is not isomorphic to the wreath product Zp ≀ Zp. Furthermore, we provide a general description of the transitive groups of degree p 2 whose Sylow p-subgroup is isomorphic to Zp ≀ Zp, and explicitly determine most of them. As applications, we solve the Cayley Isomorphism problem for Cayley objects of an abelian group of order p 2, explicitly determine the full automorphism group of Cayley graphs of abelian groups of order p 2, and find all nonnormal Cayley graphs of order p 2.