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 ρ1(G)≥ρ2(G)≥⋯≥ρn−1(G)≥ρn(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′ asσN(G,G′)=∑i=1n|ρi(G)−ρi(G′)|p,p≥1. Define the cospectrality of G bycsN(G)=min⁡{σN(G,G′):G′ not isomorphic to G}. LetcsnN=max⁡{csN(G):G a graph on n vertices}. In this paper, we give an upper bound on csN(G) in terms of the graph parameters. Moreover, we obtain an exact value of csnN. An upper bound on the distance between the normalized Laplacian spectra of two graphs has been presented in terms of Randić 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.