Characteristic polynomial of a generalized complete product of matrices

For a simple graph G, let (G) over bar denote the complement of G relative to the complete graph and let P(G) (x) = det(xI - A(G)) where A(G) denotes the adjacency matrix of G. The complete product G del H of two simple graphs G and H is the graph obtained from G and H by joining every vertex of G to every vertex of H. In [2] P(G del H) (x) is represented in terms of P(G), P((G) over bar), P(H) and P((H) over bar). In this paper we extend the notion of complete product of simple graphs to that of generalized complete product of matrices and obtain their characteristic polynomials. (C) 2010 Elsevier Inc. All rights reserved.