Product Cayley Graph of a Finite Group

Abstract

In this paper, we introduce and study a new graph associated with a finite group, called the Product Cayley graph. Let $G$ be a finite group and let $S$ be a symmetric generating set of $G$. The Product Cayley graph $\Gamma_{pc}(G,S)$ is a graph with the vertex set $G$, and two distinct vertices $x$ and $y$ are adjacent if either $(xy)x^{-1}\in S$ or $(yx)y^{-1}\in S$. We investigate the fundamental structural properties of this graph, including connectedness, regularity, automorphisms, bipartiteness, and spectral properties. Special attention is devoted to finite abelian groups, where the structure of the graph is characterized in terms of the generating set. Furthermore, we study the non-abelian case by considering the dihedral groups $D_{n}$. We first classify symmetric generating sets of small cardinality and determine the conditions under which they generate the whole dihedral group. Based on these classifications, representative families of Product Cayley graphs of dihedral groups are investigated. Several graph invariants, including degree sequence, diameter, clique number, chromatic number, adjacency spectrum, and energy, are obtained.