The Laplacian spectral ratio of connected graphs

TL;DR

This paper investigates the Laplacian spectral ratio of connected graphs, providing improved bounds, counterexamples to existing conjectures, and proposing a new conjecture related to graph spectral properties.

Contribution

It offers new bounds for the Laplacian spectral ratio, presents counterexamples to a conjecture on trees, and introduces a new conjecture in spectral graph theory.

Findings

Improved bounds for the Laplacian spectral ratio.

Counterexamples to the existing conjecture on trees.

Proposal of a new conjecture on spectral ratios.

Abstract

Let $G$ be a simple connected undirected graph. The Laplacian spectral ratio of $G$, denoted by $R_L(G)$, is defined as the quotient between the largest and second smallest Laplacian eigenvalues of $G$, which is closely related to the structural parameters of a graph (or network), such as diameter, $t$-tough, perfect matching, average density of cuts, and synchronizability, etc. In this paper, we obtain some bounds of the Laplacian spectral ratio, which improves the known results. In addition, we give counter-examples on the upper bound of the Laplacian spectral ratio conjecture of trees, and propose a new conjecture.