Spectra of some new extended corona

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.