The signless Laplacian spectral radius of graphs without trees

TL;DR

This paper establishes a sharp upper bound for the signless Laplacian spectral radius of graphs that contain no trees, characterizing the extremal graphs that achieve this bound, thus contributing to spectral extremal graph theory.

Contribution

It provides a new upper bound for the signless Laplacian spectral radius of graphs without trees and characterizes the extremal graphs attaining this bound.

Findings

Sharp upper bound for the spectral radius established

Characterization of extremal graphs achieved

Connection to Erdős-Sós conjecture highlighted

Abstract

Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of a simple graph of order $n$, where $D(G)$ and $A(G)$ are the degree diagonal matrix and the adjacency matrix of $G$, respectively. In this paper, we present a sharp upper bound for the signless spectral radius of $G$ without any tree and characterize all extremal graphs which attain the upper bound, which may be regarded as a spectral extremal version for the famous Erd\H{o}s-S\'{o}s conjecture.