Algebraic Structures and Their Applications Vol. 6 No. 1 ( 2019 ) pp 47-54. GRAPH PRODUCT OF GENERALIZED CAYLEY GRAPHS OVER POLYGROUPS D. HEIDARI * AND B. DAVVAZ Abstract. In this paper, we introduce a suitable generalization of Cayley graphs that is defined over polygroups (GCP-graph) and give some examples and properties. Then, we mention a generalization of NEPS that contains some known graph operations and apply to GCP-graphs. Finally, we prove that the product of GCP-graphs is again a GCP-graph. 1. Introduction and preliminaries A graph product is a binary operation on graphs. So, many large graphs can be constructed from existing smaller graphs. In [10], Li et al. studied the properties of the lexicographic product of vertex-transitive and of edge-transitive graphs, and of the Cayley graphs. They proved that the lexicographic product of vertex-transitive (edge-transitive) graphs is a vertex- transitive (edge-transitive) graph and, in particular, the lexicographic product of Cayley graphs is a Cayley graph. DOI: 10.29252/as.2019.1340 MSC(2010): Primary: 20N20, 05C25 Keywords: Simple graph, Caylay graph, polygroup, GCP-graph, graph product. Received:22 Nov 2018, Accepted: 17 Feb 2019. ∗Corresponding author c ⃝ 2019 Yazd University. 47