Spectral and size conditions for spanning k-trees in tough graphs

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 $\tau (G)=\min \{\frac{|S|}{c(G-S)}:S\subseteq V(G),c(G-S)>1\}$, 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 $\frac{1}{k-1}\le \tau (G)<\frac{1}{k-2}$, 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 $\frac{1}{k}\le \tau (G)<\frac{1}{k-1}$. 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.