$A_\alpha$-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.