Spectral analysis of a graph on the special set 𝒮

Let [Formula: see text] be the ring of integer modulo [Formula: see text] with two binary operators, addition [Formula: see text] and multiplication [Formula: see text], where [Formula: see text] is a positive integer. The special set [Formula: see text] is defined as [Formula: see text]. Our purpose in the present paper is to propose a new family of interconnection networks that are Cayley graphs on this special set [Formula: see text] and denote it by [Formula: see text]. In this paper, we define a relationship between [Formula: see text] and [Formula: see text], [Formula: see text] is a derived graph from [Formula: see text] by removing [Formula: see text] edges, where [Formula: see text] is a known fixed value. We also give the spectrum of absorption Cayley graph, unitary addition Cayley graph, and [Formula: see text]. We also provide values of [Formula: see text] for which the graph [Formula: see text] is hyperenergetic and discuss the structural properties of this graph, such as planarity and connectedness.