Distance spectral radius conditions for edge-disjoint spanning trees and a forest with constraints

TL;DR

This paper establishes spectral radius conditions based on the distance matrix that guarantee a graph contains $k$ edge-disjoint spanning trees and a large forest with specific properties, extending previous work.

Contribution

It introduces new spectral radius bounds that ensure the existence of complex spanning structures in graphs, generalizing prior results to more refined properties.

Findings

Spectral radius bounds imply property P(k, δ) in general graphs.

Results apply to both general and bipartite graphs with minimum degree conditions.

Extends previous work from spanning trees to structured forests with constraints.

Abstract

Let $k\ge 2$ be a positive integer and let $G$ be a simple graph of order $n$ with minimum degree $\delta$. A graph $G$ is said to have property $P(k, d)$ if it contains $k$ edge-disjoint spanning trees and an additional forest $F$ with edge number $|E(F)| > \frac{d-1}{d}(n-1)$, such that if $F$ is not a spanning tree, then $F$ has a component with at least $d$ edges. Let $D(G)$ be the distance matrix of $G$. We denote $\rho_D(G)$ as the largest eigenvalue of $D(G)$, which is called the distance spectral radius of $G$. In this paper, we investigate the relationship between the distance spectral radius and the property $P(k, \delta)$. We prove that for a connected graph $G$ of order $n \ge 2k+8$ with minimum degree $\delta \ge k+2$, if $\rho_D(G) \le \rho_D(K_{k-1} \vee (K_{n-k} \cup K_1))$, then $G$ possesses property $P(k, \delta)$. Furthermore, for a connected balanced bipartite graph $G$ of order $n \ge 4k+8$ with minimum degree $\delta \ge k+2$, we show that if $\rho_D(G) \le \rho_D(K_{\frac{n}{2}, \frac{n}{2}} \setminus E(K_{1, \frac{n}{2}-k+1}))$, then $G$ also possesses property $P(k, \delta)$. Our results generalize the work of Fan et al. [Discrete Appl. Math. 376 (2025), 31--40] from the existence of $k$ edge-disjoint spanning trees to the more refined structural property $P(k, \delta)$.