Some Of The Results Of The(Signless) Laplace Spectrum Of The Graph

Suppose G,H1,...,H,be m+1 not intersecting simple graphs,where m=|E(G)|.The generalized edge corona,denoted G(?)Hi the graph obtained by joining two end-vertices of the i-th edge of G to every vertex of Hi.In the second part of this paper,we determine and study the characteristic,Laplacian and signless Laplacian polynomial of(?)Hi.This leads us to construct new pairs of cospectral,L-cospectral and Q-cospectral graphs.let Y=(Y1,Y2,···,Yn)T?Rn,then(y1p+y2p+…ynp)1/p-||Y|| is the p-norm of Y.If ||Y||=1,then Y is the p-normalized.Suppose M be non-negative irreducible matrix.You know by the Perron-Frobenius theorem that for any given 1?p??,the matrix M has a unique,positive p-normalized principal eigenvector Y corresponding to the spectral radius of M,then Y is called the principal eigenvector of the matrix M.In the third part of this paper,we determine upper and lower bounds on the minimal and maximal entries of the principal eigenvector of the signless Laplacian matrix.The Laplacian matrix L(G)is positive semidefinite,and its maximum eigenvalue is not always simple.Suppose that X=(x1,x2,...,xn)T is the p-normalized eigenvector corresponding to the spectral radius of L(G).In addition,we also determine lower bounds on the maximal entry of the vector X=(|x1|,|x2|,…,|xn|)T.