Characteristic polynomial of a generalized complete product of matrices

For a simple graph G, let G¯ denote the complement of G relative to the complete graph and let PG(x)=det(xI-A(G)) where A(G) denotes the adjacency matrix of G. The complete product G∇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] PG∇H(x) is represented in terms of PG, PG¯, PH and PH¯. 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.