Girth and Laplacian eigenvalue distribution

TL;DR

This paper investigates the distribution of Laplacian eigenvalues in connected graphs based on girth, establishing bounds and characterizing extremal graphs for specific girth and eigenvalue count conditions.

Contribution

It provides new bounds on the number of Laplacian eigenvalues within certain intervals related to girth and characterizes extremal graphs achieving these bounds.

Findings

For graphs with girth at least 4, the number of eigenvalues in specified intervals is bounded by n-g.

Graphs achieving the bounds for k=1,2 are explicitly characterized.

The paper classifies graphs with girth 3 where the eigenvalue count reaches n-1, n-2, n-3.

Abstract

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 \[ n(G,k)\le n-g. \] Those graphs achieving the bound for $k=1,2$ are determined. We also determine the graphs $G$ with $g=3$ such that $n(G,k)=n-1, n-2, n-3$.