Spanning trees and signless Laplacian spectral radius in graphs

TL;DR

This paper explores the relationship between spanning trees and the signless Laplacian spectral radius of graphs, providing conditions that guarantee the existence of spanning trees with bounded leaf degree, and establishing sharp bounds through extremal graph constructions.

Contribution

It introduces a spectral condition based on the signless Laplacian spectral radius that ensures a graph contains a spanning tree with a specified maximum leaf degree.

Findings

Derived a sufficient spectral condition for spanning trees with bounded leaf degree.

Established sharp bounds for the signless Laplacian spectral radius related to spanning trees.

Constructed extremal graphs to demonstrate the bounds are tight.

Abstract

Let $G$ be a connected graph and let $k$ be a positive integer. Let $T$ be a spanning tree of $G$. The leaf degree of a vertex $v\in V(T)$ is defined as the number of leaves adjacent to $v$ in $T$. The leaf degree of $T$ is the maximum leaf degree among all the vertices of $T$. Let $A(G)$ be the adjacency matrix of $G$ and $D(G)$ be the diagonal degree matrix of $G$. Let $Q(G)=D(G)+A(G)$ be the signless Laplacian matrix of $G$. The largest eigenvalue of $Q(G)$, denoted by $q(G)$, is called the signless Laplacian spectral radius of $G$. In this paper, we investigate the connection between the spanning tree and the signless Laplacian spectral radius of $G$, and put forward a sufficient condition based upon the signless Laplacian spectral radius to guarantee that a graph $G$ contains a spanning tree with leaf degree at most $k$. Finally, we construct some extremal graphs to claim all the bounds obtained in this paper are sharp.