ITALIAN JOURNAL OF PURE AND APPLIED MATHEMATICS – N.39–2018 (628–635) 628 NORMAL EDGE-TRANSITIVE CAYLEY GRAPHS WHOSE ORDER ARE A PRODUCT OF THREE PRIMES Modjtaba Ghorbani ∗ Mahin Songhori Mina Rajabi Parsa Department of Mathematics Faculty of Science Shahid Rajaee Teacher Training University Tehran, 16785-136 I. R. Iran mghorbani@srttu.edu Abstract. The Cayley graph X = Cay(G, S) on group G with respect to connection set S is normal edge-transitive, if N Aut(X) (R(G)) acts transitively on edge set. In this paper we determine the structure of automorphism group of all tetravalent normal edge-transitive Cayley graphs of order pqr. Keywords: wreath product, normal edge-transitive Cayley graph, automorphism group. 1. Introduction All graphs considered here are finite, simple, undirected and connected. The vertex set and the edge set of graph Γ are denoted by V (Γ) and E(Γ), respec- tively. An automorphism of graph Γ is a mapping from the vertices of Γ back to vertices of Γ such that the resulting graph is isomorphic with Γ. The set of all automorphisms of Γ under the composition of mappings forms a group known as the automorphism group denoted by Aut(Γ). Among all graphs, determin- ing the automorphism group of Cayley graphs is a very difficult task. Thus, the majority of this research has been focused on normal edge-transitive Cayley graphs. Due to the complexity, this article only briefly examines properties of Cayley graphs of order pqr, where p, q, r are prime numbers. Here, in the next section, we give the necessary definitions and some pre- liminary results. Section three contains main results of this paper and the automorphism group of Cayley graphs of order pqr is verified in this section. *. Corresponding author