On Isomorphisms Of The Cayley Graphs Of Small Valencies

Let G be a finite group and let S be a subset of G such that 1(?) S and S is symmetric, that is, S^-1={s^-1 s∈S} is equal to S. The Cayley graph Cay(G, S) on G with respect to S is defined as the graph with vertex set G and edge set {{g, h} | g, h∈G, gh^-1∈S}. If for any subset S of G not containing the identity element 1, Cay(G, S) is isomorphic to Cay(G, T) if and only if S^α=T for some a∈Ant(G), then S is called a CI-subset of G, and Cay(G, S) is called a CI-graph. For a positive integer m, G is called an m-CI-group provided that every subset, say S, of G satisfying S^-1=S and |S|≤m is a CI-subset, and if every generating subset S of G satisfying S^-1=S and |S|≤m is a CI-subset, then G is called a weakly m-CI-group. In particular, a |G|-CI-group G is called a CI-group.The CI-property of Cayley graphs is one of the important problems in the studying of the isomorphism of Cayley graphs. In the literature, there are a lot of work have been done on this problem and in this paper, we shall consider the weakly m-CI-property for some classes of groups with m=3 or 4. Let p be an odd prime. First, it is proved that every group of order 2p² is a weakly 3-CI-group. As an application, all connected cubic Cayley graphs of order 2p² are classified. Second, let. G be a group of order 4p and S a generating subset of Gsuchthat 1 (?) S, S^-1=S and |S|≤3. It is shown that the Cayley graph Cay(G,S) is non-CI if and only if G=〈a, b|a^2p=b²=1, b^-1ab=a⁽-1〉and S^α={b, a, a^-1}, {b, ba, ba^-1} or {b, a^P, a²b} for some a∈Ant(G). Finally, we prove that every connected tetravalent Cayley graph of the generalized quaternion group Q_4n=〈a, b| a²n =1, b²=aⁿ, b^-1ab=a^-1〉(n≥2) is isomorphic to Cay(Q_4n, {a, a^-1, b, b^-1})or Cay(Q_4n, {b, b^-1, ab, (ab)^-1}). Furthermore, if n=2 then these two graphs are isomorphic to the complete bipartite graph K_4,4 and if n＞2, then they are non-isomorphic. Since Q_4n cannot be generated by an element or an involution together with an element of order greater than 2, it follows that Q_4n is a weakly 4-CI-group.