integral graphs

For two graphs G
and H with n and m vertices, the corona G∘H of G and H is the graph obtained by taking one copy of G and n copies of H and then joining the ith vertex of G to every vertex in the ith copy of H. The neighborhood corona G⋆H of G and H is the graph obtained by taking one copy of G and n copies of H and joining every neighbor of the ith vertex of G to every vertex in the ith copy of H. In this paper, we define four new extensions of corona and neighborhood corona of two graphs G and H; named the identity-extended corona, identity-extended neighborhood corona, neighborhood extended corona and neighborhood extended neighborhood corona and then determine the spectrum of their adjacency matrix, where H
is a regular graph. As an application, we exhibit infinite families of integral graphs.

In this paper, we define duplication corona, duplication neighborhood corona and duplication edge corona of two graphs. We compute their adjacency spectrum, Laplacian spectrum and signless Laplacian. As an application, our results enable us to construct infinitely many pairs of cospectral graphs and also integral graphs.