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)≥⋯≥ρ_n−1(G)≥ρ_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 σ_N(G,G^′)=∑i=1n|ρ_i(G)−ρ_i(G^′)|^p,p≥1. Define the cospectrality of G by cs^N(G)=min⁡{σ_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 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.