Study on adjacent spectrum of two kinds of joins of graphs

The multiple subdivision graph of a graph G, denoted by S-n(G), is the graph obtained by inserting n paths of length 2 replacing every edge of G. When n = 1, S-1(G) = S(G) is the subdivision graph of G. Let G(1) be a graph with n(1) vertices and m(1) edges, G(2) be a graph with n(2) vertices and m(2) edges. The quasi-corona SG-vertex join G(1) Delta G(2) of G(1) and G(2) is the graph obtained from S(G(1)) boolean OR G(1) and n(1) copies of G(2) by joining every vertex of G(1) to every vertex of G(2), and multiple SG-vertex join G(1) circle dot G(2) is the graph obtained from S-n(G(1)) boolean OR G(1) and G(2) by joining every vertex of G(1) to every vertex of G(2). In this paper, we calculate analytic expression of characteristic polynomial of adjacency matrix of the above two types of joins of graphs for the case of G(1) being a regular graph. Then we obtain their adjacency spectra for the case of G(1) and G(2) being regular graphs.