The normalized Laplacian spectra of the corona and edge corona of two graphs

Let G(1) and G(2) be two simple graphs. Denote the corona and the edge corona of G(1) and G(2) by G(1) circle G2 and G1 (sic) G2, respectively. In this paper, first, the normalized Laplacian spectrum of G1 circle G2 is given in terms of that of G1 and G2 when both G1 and G2 are regular. Then the normalized Laplacian spectrum of G1 (sic) G2 is given in terms of that of G1 and G2 for any connected graph G1 and a regular graph G2. Finally, as applications, the number of spanning trees and the degree Kirchhoff index of G1 circle G2 and G1 (sic) G2 are considered.