Spectra Of The Generalized Edge Corona Of Graphs And Its Applications

Let G and H₁,H₂,···,H_mbe simple graphs,where G has m edges.Taking one copy of graphs G and H₁,H₂,···,H_m,and then joining two end-vertices of the i'th edge e_iof G to each vertex of H_i,for each i?{1,2,···,m},and obtain a new graph,denoted by G[H_i]₁^m,which is called to be the generalized edge corona of graph G and H₁,H₂,···,H_m.Delete all the edges of graph G from the generalized edge corona of graph,and obtain a new graph,which is called to be the modified generalized edge corona of graph,denoted by ???.In this paper,we study the characteristic polynomial,Laplacian polynomial,signless Laplacian polynomial and normalized Laplacian polynomial of G[H_i]₁^m and ???,and then the corresponding spectra is obtained from their characteristic polynomial,such as the spectrum,Laplacian spectrum,signless Laplacian spectrum and the normalized Laplacian spectrum of G[H_i]₁^m and ???.As applications,we also count the Kirchhoff index,degree-Kirchhoff index and the number of spanning trees of G[H_i]₁^m and ???.