The signless Laplacian spectral radius of subgraphs of regular graphs

TL;DR

This paper establishes new bounds on the signless Laplacian spectral radius of subgraphs of regular graphs, depending on their connectivity and diameter, providing insights into spectral properties related to graph structure.

Contribution

The paper introduces two novel bounds for the spectral radius of subgraphs of regular graphs, considering their connectivity and diameter, and compares their effectiveness.

Findings

Derived bounds depend on graph connectivity and diameter.

Compared bounds show which is tighter under specific conditions.

Provides theoretical insights into spectral properties of regular graphs.

Abstract

Let $q(H)$ be the signless Laplacian spectral radius of a graph $H$. In this paper, we prove that \\1. Let $H$ be a proper subgraph of a $\Delta$-regular graph $G$ with $n$ vertices and diameter $D$. Then $$2\Delta - q(H)>\frac{1}{n(D-\frac{1}{4})}.$$ \\2. Let $H$ be a proper subgraph of a $k$-connected $\Delta$-regular graph $G$ with $n$ vertices, where $k\geq 2$. Then $$2\Delta-q(H)>\frac{2(k-1)^{2}}{2(n-\Delta)(n-\Delta+2k-4)+(n+1)(k-1)^{2}}.$$ Finally, we compare the two bounds. We obtain that when $k>2\sqrt{\frac{(n-\Delta)(n+\Delta-4)}{n(4D-3)-2}}+1$, the second bound is always better than the first. On the other hand, when $k<\frac{2(n-\Delta)}{\sqrt{n(4D-3)-2}}+1$, the first bound is always better than the second.