The number of Laplacian eigenvalues of a graph in interval [3,n].

Question

There are several upper and lower bounds for $m_G[2,n]$ (the number of Laplacian eigenvalues of a graph $G$ with $n$ vertices in the interval $[2,n]$). I want to know whether there exists any bound if we replace $k\in \mathbb{N}$ instead of $2$. Especially when the interval is $[3,n]$. At least is there any bound if the graph is tree or other special cases?

Answer 1

We can trivially say that the sum of the eigenvalues is twice the number of edges, which constrains things somewhat. For example, a tree has $n-1$ nonzero eigenvalues summing to $2(n-1)$, so their average value is $2$. At most $\frac23(n-1)$ of them can be in the interval $[3,n]$.

In a dense graph, this gives us a lower bound instead. If a graph has $\alpha \binom{n}{2}$ edges for some $\alpha \in [0,1]$, the average of the $n-1$ largest eigenvalues is $\alpha n$, so for any $\beta < \alpha$, at least $\frac{\alpha-\beta}{1-\beta}(n-1)$ of them must be in the interval $(\beta n,n]$.