Let $ℤ_{n}$ be the ring of integer modulo $n$ with two binary operators, addition $(+)$ and multiplication $(.)$ , where $n$ is a positive integer. The special set $𝒮$ is defined as $𝒮 = {a \in ℤ_{n} : (\exists b \in ℤ_{n}) b a \equiv a, b ≢ a, b ≢ 1}$ . Our purpose in the present paper is to propose a new family of interconnection networks that are Cayley graphs on this special set $𝒮$ and denote it by $Ω^{-} (Z_{n})$ . In this paper, we define a relationship between $G$ and $G_{e}^{*}$ , $G_{e}^{*}$ is a derived graph from $G$ by removing $r$ edges, where $r$ is a known fixed value. We also give the spectrum of absorption Cayley graph, unitary addition Cayley graph, and $Ω^{-} (Z_{n})$ . We also provide values of $n$ for which the graph $Ω^{-} (ℤ_{n})$ is hyperenergetic and discuss the structural properties of this graph, such as planarity and connectedness.

Communicated by Xiao-Dong Zhang

Keywords:

AMSC: 05C22, 05C75

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