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Abstract:The toughness of a graph is a crucial parameter for characterizing its structural properties. The toughness of a non-complete graph $G$ is defined as $\tau(G) = \min \{ \dfrac{|S|}{c(G - S)} : S \subseteq V(G), c(G-S) > 1 \}$, where $c(G)$ denotes the number of components of $G$. We define $\tau(K_n) = \infty$. A graph $G$ is said to be $\tau$-tough if $|S| \ge \tau \cdot c(G-S)$ for every vertex cut $S$ of $G$. Let $k \ge 3$ be an integer. For $\frac{1}{k-\eta}$-tough graphs with $\eta \in \{0, 1\}$, Liu, Fan and Shu \cite{a34} derived sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Jia and Lu \cite{a24}, for the case $\frac{1}{k-1} \leq \tau(G) < \frac{1}{k-2}$, established sufficient conditions in terms of the spectral radius and the signless Laplacian spectral radius for the existence of a spanning $k$-tree. Motivated by these results, in this paper, we further investigate sufficient conditions for the existence of a spanning $k$-tree when $\frac{1}{k} \leq \tau(G) < \frac{1}{k-1}$. Specifically, for a connected $\frac{t}{t(k-1)+1}$-tough graph of sufficiently large order $n$ (where $t \ge 1$ is an integer), we provide sufficient conditions for the existence of a spanning $k$-tree in terms of the spectral radius and the signless Laplacian spectral radius. Furthermore, we establish a lower bound on the size (number of edges) to guarantee the existence of a spanning $k$-tree.

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From: Tao Tian [view email]
[v1] Thu, 18 Jun 2026 14:33:42 UTC (207 KB)