THE Aα-SPECTRUM OF GRAPH PRODUCT

Let A(G) and D(G) denote the adjacency matrix and the diagonal matrix of vertex degrees of G, respectively. Define A(alpha)(G) = alpha D(G) + (1-alpha)A(G) for any real alpha is an element of [0, 1]. The collection of eigenvalues of A(alpha)(G) together with multiplicities is called the A(alpha)-spectrum of G. Let G square H, G[H], G x H and G circle plus 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 A, v-spectrum of G square H for arbitrary graphs G and H, and G[H] for arbitrary graph G and regular graph H is given. Furthermore, A(alpha)-spectrum of the generalized lexicographic product G[H-1, H-2 ,..., H-n] d for n-vertex graph G and regular graphs 1/,'s is considered. At last, the spectral radii of A(alpha)(G x H) and A(alpha)(G circle plus H) for arbitrary graph G and regular graph H are given.