Laplacain eigenvalue distribution and diameter of graphs

TL;DR

This paper investigates the distribution of Laplacian eigenvalues in relation to the diameter of connected graphs, providing bounds and characterizations for eigenvalue counts within specific intervals.

Contribution

It improves existing bounds on the number of Laplacian eigenvalues in certain intervals for graphs with given diameters and characterizes graphs achieving these bounds.

Findings

At most n-d+1 Laplacian eigenvalues in [n-d+1, n] for graphs with diameter d

Refined bounds on eigenvalue counts in relation to graph diameter

Characterization of graphs with extremal eigenvalue distributions

Abstract

Let $G$ be a connected graph on $n$ vertices with diameter $d$. It is known that if $2\le d\le n-2$, there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+2, n]$. In this paper, we show that if $1\le d\le n-3$, there are at most $n-d+1$ Laplacian eigenvalues in the interval $[n-d+1, n]$. Moreover, we try to identify the connected graphs on $n$ vertices with diameter $d$, where $2\le d\le n-3$, such that there are at most $n-d$ Laplacian eigenvalues in the interval $[n-d+1, n]$.