The Aα-spectrum of graph product

Abstract

Let A(G) and D(G) denote the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. Define (formula presented) for any real α [0, 1]. The collection of eigenvalues of Aα(G) together with multiplicities is called the Aα-spectrum of G. Let G H, G[H], G × H and G H be the Cartesian product, lexicographic product, directed product and strong product of graphs G and H, respectively. In this paper, a complete characterization of the Aα-spectrum of G H for arbitrary graphs G and H, and G[H] for arbitrary graph G and regular graph H is given. Furthermore, Aα-spectrum of the generalized lexicographic product G[H1, H2, . . ., Hn] for n-vertex graph G and regular graphs Hi ’s is considered. At last, the spectral radii of Aα(G × H) and Aα(G H) for arbitrary graph G and regular graph H are given.