Some Results On The Normalized Laplacian Polynomial Of A Graph

Let G1,G2 be two simple graphs.Denote the join graph,the corona and the edge corona of G1 and G2.In the second chapter of this paper,first,we obtain an expression of the Normalized Laplacian polynomial for the join graph of G1 and G2.Then based on this expression,we obtain the relation between the Normalized Laplacian eigenvalues of the join graph and those of G1 and G2.Then,the Normalized Laplacian spectrum of the corona is given in terms of that of G1 and G2when both G1 and G2are regular,and the Normalized Laplacian spectrum of the edge corona is given in terms of that of G1 and G2for any connected graph G1 and a regular graph G2.In the third chapter,we first give a combinatoric expression for the coefficients of the Normalized Laplacian polynomial in terms of elementary subgraphs of the graph.Then based on the expression,we prove a series of relations between the Normalized Laplacian spectrum and the structure of the graph.Finally,as applications,we compute the number of spanning trees and the degree Kirchhoff Index of several kinds of graphs.