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.
Zhen et al. (Tue,) studied this question.