A study on $A_\alpha$-spectrum and $A_\alpha$-energy of unitary addition Cayley graphs

Abstract

The unitary addition Cayley graph $G_n$, $n\in Z^+$ is the graph whose vertex set is $Z_n$, the ring of integers modulo $n$ and two vertices $u$ and $v$ are adjacent if and only if $u + v \in \cup_n$ where $\cup_n$ is the set of all units of the ring. The $A_\alpha$-matrix of a graph $G$ is defined as $A_\alpha (G) = \alpha D(G) + (1-\alpha)A(G)$, $\alpha \in [0, 1]$, where $D(G)$ is the diagonal matrix of vertex degrees and $A(G)$ is the adjacency matrix of $G$. In this paper, we investigate the $A_\alpha$-eigenvalues for unitary addition Cayley graph and its complement. We determine bounds for $A_\alpha$-eigenvalues of unitary addition Cayley graph when its order is odd. Consequently, we compute the $A_\alpha$-energy of both $G_n$ and its complement, $\overline{G_n}$, for $n={p}^m$ where ${p}$ is a prime number and $n$ even. Moreover, we obtain some bounds for energies of $G_n$ and $\overline{G}_n$ when $n$ is odd. We also define $A_\alpha$-borderenergetic and $A_\alpha$-hyperenergetic graphs and observe some classes for each.