$A_{\alpha }$-energy of graphs formed by some unary operations

Abstract

Let $G$ be a graph on $p$ vertices with adjacency matrix $A(G)$ and degree matrix $D(G)$. For each $\alpha \in [0,1]$, the $A_{\alpha }$-matrix is defined as $A_{\alpha }(G)=\alpha D(G)+(1-\alpha )A(G)$. In this paper, we compute the $A_{\alpha }$-characteristic polynomial, $A_{\alpha }$-spectra and $A_{\alpha }$-energy of some non-regular graphs obtained from unary operations on graphs like middle graph, central graph, m-splitting, and closed splitting graph. Also, we determine the $A_{\alpha }$-energy of regular graphs like m-shadow, closed shadow, extended bipartite double graph, iterated line graph and m-duplicate graph. Furthermore, we identified some graphs that are $A_{\alpha }$-equieneregetic and $A_{\alpha }$-borderenergetic.