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

The paper advances the theory of spanning trees with constrained leaf properties by deriving size- and spectral-radius criteria that guarantee a spanning tree with leaf distance at least $d$ and, separately, a spanning tree with leaf degree at most $k$, all relative to the graph's minimum degree. It introduces extremal constructions like $H_l=K_{l(d-2)}\vee(K_{n-ld}\cup2lK_1)$ and various $K_t$-based joins to tightly bound the relevant spectra ($\rho$ and $q$) and sizes, and then proves a sequence of sharp theorems (tt1--tt6 and ta--tc) showing that exceeding these bounds forces the desired spanning trees, with equality cases and isomorphism-exclusions precisely characterized. The methods blend equitable partition analysis, extremal graphs, and polynomial root arguments to connect combinatorial structure with spectral information, yielding improvements over prior work such as Ao, Liu, Yuan, Ng and Cheng (2023) and related results on Kaneko’s conjecture variants. Overall, the results provide a comprehensive spectral- and size-based toolkit for guaranteeing leaf-structure properties in spanning trees under minimum-degree conditions, with clear implications for graph design and spectral 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).

Theorems & Definitions (19)

• Definition 2.1
• Lemma 2.2
• Lemma 2.3
• Conjecture 3.1
• Theorem 3.2
• Theorem 3.4
• Theorem 3.8
• Theorem 3.9
• Theorem 3.10
• Theorem 3.11