Energy of central and middle graph of a regular graph

Abstract

An eigenvalue of a graph $G$ is the eigenvalue of its adjacency matrix. The energy $E(G)$ of $G$ is the sum of the absolute values of its eigenvalues.  Two graphs having same energy and same number of vertices are called equienergetic  graphs. If ${\mu }_1, \ldots, {\mu }_n$ are the eigenvalues of the adjacency matrix of $G$ and ${\theta }_1, \dots, {\theta }_n$ are the eigenvalues of the adjacency matrix of all-one matrix $(J_n$), then the energy of the Central graph of a simple, connected, $r$-regular graph $E(C(G))$ and the energy of a Middle graph of a simple, connected, 2-regular graph $E(M(G))$ is derived in terms of the eigenvalues of the original graph $G$ by us in this paper.