Laplacian eigenvalue distribution of a graph with given independence number

Abstract

For a graph G, let α(G) be the independence number of G, let L(G) be the Laplacian matrix of G, and let mGI be the number of eigenvalues of L(G) in the interval I. Ahanjideh, Akbari, Fakharan and Trevisan proved that α(G)≤mG[0,n−α(G)] if G is an n-vertex connected graph. Choi, Moon and Park characterized graphs with α(G)=mG[0,n−α(G)] for α(G)=2 and α(G)=n−2. In this paper, we give a characterization for α(G)=3 and α(G)=n−3.