On the Extended Adjacency Eigenvalues of Graphs and Applications

Let Aex(G) be the extended adjacency matrix of G. The eigenvalues of Aex(G) are called extended adjacency eigenvalues of G. The sum of the absolute values of eigenvalues of the Aex-matrix is called the extended adjacency energy Eex(G) of G. In this paper, we obtain the Aex-spectrum of the joined union of regular graphs in terms of their adjacency spectrum and the eigenvalues of an auxiliary matrix. Consequently, we derive the Aex-spectrum of the join of two regular graphs, the lexicographic product of regular graphs, and the Aex-spectrum of various families of graphs. Further, as applications of our results, we construct infinite classes of infinite families of extended adjacency equienergetic graphs. We show that the Aex-energy of the join of two regular graphs is greater than or equal to their energy. We also determine the Aex-eigenvalues of the power graph of finite abelian groups.