The spectrum and the signless Laplacian spectrum of coronae

Let G1,G2 be two simple connected graphs. Denote the corona and the edge corona of G1,G2 by G1∘G2 and , respectively. In this paper, we first introduce a new invariant, the M-coronal of a graph matrix M, where the matrix M is associated with a graph in a prescribed way. Then it is used to compute the signless Laplacian spectrum of G1∘G2 and in terms of the signless Laplacian spectrum of G1 and G2. In addition, the spectrum of are also given in terms of the spectrum of G1 and G2. Finally, as an application of these results, we construct many pairs of nonisomorphic signless Laplacian cospectral graphs.