On the spectra of some graphs

Let G1,G2, …, Gk be simple graphs and xi be an arbitrary vertex of Gi, i = 1, 2, …, k. For any simple graph H with vertex set {x1, x2, …, xk}, the graph G obtained by the union G1, G2, …, Gk and H is denoted by K = H(G1,G2, …, Gk). In this paper we calculate characteristic polynomial of adjacency matrix of this graph in terms of the characteristic polynomial of the graphs Gi and H. We also consider some special cases for the graphs Gi and H, and calculate spectra of K.