Laplacian eigenvalue distribution and girth of graphs

Let G be a connected graph on n vertices with girth g. Let mGI denote the number of Laplacian eigenvalues of graph G in an interval I. In this paper, we show that if G is not a cycle, then mG(n-g+3,n]≤ n-g. Moreover, we prove that mG(n-g+3,n]= n-g if and only if G C₃ or G K3,2 or G U₁, where U₁ is obtained from a cycle by joining a single vertex with a vertex of this cycle.