Girth and Laplacian eigenvalue distribution

Let $G$ be a connected graph of order $n$ with girth $g$. For $k=1,\dots ,\min \{g-1,n-g\}$, let $n(G,k)$ be the number of Laplacian eigenvalues (counting multiplicities) of $G$ that fall inside the interval $[n-g-k+4,n]$. We prove that if $g\ge 4$, then $$n(G,k)\le n-g\mathrm{.}$$ Those graphs achieving the bound for $k=1,2$ are determined. We also determine the graphs $G$ with $g=3$ such that $n(G,k)=n-1,n-2,n-3$.