Classification of graphs by Laplacian eigenvalue distribution and independence number

Abstract

Let (Formula presented.) denote the number of Laplacian eigenvalues of a graph G in an interval I and let (Formula presented.) denote the independence number of G. In this paper, we determine the classes of graphs that satisfy the condition (Formula presented.) when (Formula presented.) and (Formula presented.), where n is the order of G. When (Formula presented.), (Formula presented.) for some (Formula presented.). When (Formula presented.), there are two types of graphs (Formula presented.) and (Formula presented.) of order n = p + q + r + 2, which we call the binary star graphs. Also, we show that the binary star graphs with p = r are determined by their Laplacian spectra.