On the spectra of quasi join of graphs and families of integral graphs

The subdivision graph S(G) of a graph G is formed by adding a new vertex into every edge of G. The quasi-corona subdivision-vertex join of two graphs G and G′ is a graph derived from S(G) and G′ by choosing a copy of S(G) and n1 copies of G′ and then connecting those vertices of S(G) which were in G to all the vertices of G′ . The quasicorona subdivision-edge join of two graphs G and G′ is a graph derived from S(G) and G′ by choosing a copy of S(G) and n1 copies of G′ and then connecting those vertices of S(G) which were not in G to all the vertices of G′. The adjacency, Laplacian and signless Laplacian spectrum of these graphs are determined. As a consequence, we obtain some families of integral graphs, infinite families of cospectral graphs, the number of spanning trees and the Kirchhoff index.