Isomorphism problem for Cayley graphs of Zp3

Isomorphisms of Cayley digraphs of Abelian groups

“…Suppose that 5 contains no coset of a nontrivial subgroup of G. Then by Theorem 3.3, A\ is faithful on 5 and it follows that (|G|, \A X \) divides p, and therefore, by Proposition 3.4,C.H. Li [6] of G, so we may write 5 = (c)bU {a} for some a,b,c e G with (c) = Z p . In particular, G is of rank at most 3.…”

“…Babai and Frankl in [4] investigated isomorphisms of undirected Cayley graphs of odd order and posed a conjecture that all undirected Cayley graphs of elementary Abelian groups Z^ were Cl-graphs. The conjecture has been proved for the case d = 2 by Godsil [11] and for the case d = 3 by Dobson [6]. It is actually proved that Zp for d ^ 3 are DCI-groups.…”

“…However, this is still a very difficult problem. For example, it is even not known whether Z* with p a prime are m-DCI-groups for arbitrary m, see for example [6]. One of the main aims of this paper is to give a reduction for Problem 1.1.…”

The Isomorphism Problem for Cayley Graphs on the Generalized Dicyclic Group

“…Godsil [7] proved that 2 p Z is a CI-group with respect to graphs for p a prime. Babai [3] showed that the nonabelian group of order 2p is a CI-group with respect to graphs, and the author [4] showed that the nonabelian group of order p pq, and q distinct primes, is a CI-group with respect to graphs if and only if 2 = q or 3. While some other results on the above problem are known, other than the above mentioned results of Muzychuk there are no known CI-groups with respect to graphs where the order of the group has more than three prime factors.…”

On Solvable Groups and Circulant Graphs

“…A circulant graph of order n is a graph whose automorphism group contains an n-cycle. LEMMA 2 (DOBSON, [3]). Let G be as in Definition 3, and α ∈ N be such that |α| = p.…”