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.