Distance between the normalized Laplacian spectra of two graphs

Let G = (V, E) be a simple graph of order n. The normalized Laplacian eigenvalues of graph G are denoted by rho(1)(G)>= rho 2(G) >= ... >= rho(n-1)(G) >= rho(n)(G) = 0. Also let G and G' be two nonisomorphic graphs on n vertices. Define the distance between the normalized Laplacian spectra of G and G' as sigma(N) (G, G') =Sigma(n)(i=1) vertical bar rho(i)(G) - rho(i)(G)vertical bar(p), p >= 1. Define the cospectrality of G by cs(N)(G) = min{sigma(N)(G, G') : G' not isomorphic to G}. Let cs(n)(N) = max{cs(N)(G) : G a graph on n vertices}. In this paper, we give an upper bound on cs(N)(G) in terms of the graph parameters. Moreover, we obtain an exact value of cs(N)(n). An upper bound on the distance between the normalized Laplacian spectra of two graphs has been presented in terms of Rancho energy. As an application, we determine the class of graphs, which are lying closer to the complete bipartite graph than to the complete graph regarding the distance of normalized Laplacian spectra. (C) 2017 Elsevier Inc. All rights reserved.