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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 Randić matrix of G , denoted by R, is the n× n matrix whose (i, j)−entry is 1 √ did j 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 n× n identity matrix. In this paper, by using an upper bound on the maximum modulus of the subdominant Randić eigenvalues 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 Randić eigenvalues and, from this bound, we improve the previous upper bound on the largest eigenvalue of L . Mathematics subject classification (2010): 05C50, 15A48.

A new upper bound on the largest normalized Laplacian eigenvalue