The existence of a spanning tree with leaf distance at least $d$ and leaf degree at most $k$ via the size or the spectral radius with respect to the minimum degree

TL;DR

This paper establishes new bounds based on size and spectral radius to ensure the existence of spanning trees with specified leaf distance and leaf degree constraints in connected graphs, advancing previous conjectures and results.

Contribution

It provides the first lower bounds using size and spectral radius for spanning trees with large leaf distance, and tight conditions for leaf degree constraints, improving prior work.

Findings

Lower bounds on size and spectral radius guarantee spanning trees with leaf distance at least d.

Tight conditions on size and spectral radius ensure spanning trees with leaf degree at most k.

Results extend and refine previous conjectures and partial results in graph theory.

Abstract

Let $k$, $d$ be a positive integer, $G$ be a connected graph of order $n$, $T$ be a tree. The leaf distance of a tree is defined as the minimum distance between any two leaves. For $v\in V(T)$, the leaf degree of $v$ in $T$ is the number of leaves adjacent to $v$, and the leaf degree of $T$ is defined as maximum leaf degree among the vertices of $T$. In this paper, motivated by the conjecture proposed by Kaneko (2001) and its subsequent partial confirmation by Erbes, Molla, Mousley and Santana (2017), we obtain lower bounds in terms of the size and the adjacent spectral radius to guarantee that $G$ contains a spanning tree with leaf distance at least $d$, where $4\leq d \leq n-1$. Furthermore, we obtain some tight conditions in $G$ for its size and spectral radius to ensure that $G$ has a spanning tree with leaf degree at most $k$, which improves the result of Ao, Liu, Yuan, Ng and Cheng (2023).