A new upper bound on the largest normalized Laplacian eigenvalue

Abstract

Let G be a simple undirected connected graph on n vertices. Suppose that the vertices of G are labelled 1,2,..., n. Let di be the degree of the vertex i. The Randićmatrix of G, denoted by R, is the nxn matrix whose (i,j) - entry if the vertices i and j are adjacent and 0 otherwise. The normalized Laplacian matrix of G is L = I - R, where I is the nxn identity matrix. In this paper, by using an upper bound on the maximum modulus of the subdominant Randićeigenvalues of G, we obtain an upper bound on the largest eigenvalue of L. We also obtain an upper bound on the largest modulus of the negative Randićeigenvalues and, from this bound, we improve the previous upper bound on the largest eigenvalue of L.