Spectra Of Two Kinds Variant Corona Graph

Let G1, G2 be two simple connected graphs. The partially subdivision neighbourhood corona of G1 and G2, denoted by G1/G2, is obtained by taking one copy of G1 and |V(G1)| copies of G2, and joining the neighbours of the i-th vertex of G1 to every vertex in the i-th copy of G2, then insert a new vertex into every edge of G1. The total corona of G1 and G2, denoted by G1 (?)G2, is obtained by taking one copy of T(G1) and|V(G1)| copies of G2, and joining the ith vertex of G1 to every vertex in the ith copy of G2. In this paper, we determine the adjacency spectrum, Laplacian spectrum and signless Laplacian spectrum of these two kinds of variant graphs. In addition, as many applications of these results, we consider to construct infinitely pairs of cospectral, Laplacian cospectral and signless Laplacian cospectral graphs. Moreover, we also compute the number of spanning trees of G1/G2 and G1(?) G2 in terms of the Laplacian spectra of two factor graphs G1 and G2.