Laplacian eigenvalue distribution for unicyclic graphs

Abstract

Let G be a graph and let mG[0,1) denote the number of Laplacian eigenvalues of G in the interval [0,1). For a tree T with diameter d, Guo, Xue, and Liu proved that mT[0,1)≥(d+1)/3. In this paper, we provide a lower bound for mG[0,1) when G is a unicyclic graph, in terms of the diameter and girth of G. Moreover, for the lollipop graph, under certain conditions on its diameter and girth, we give a formula for the exact value of mG[0,1).