Spectral radius and spanning tree with bounded total $k$-excess of graphs

Abstract

The binding number $b(G)$ of a graph $G$ is the minimum value of $|N_G(X)|/|X|$ taken over all non-empty subsets $X$ of $V(G)$ such that $N_G(X)\neq V(G)$. A graph $G$ is called $1$-binding if $b(G)\geq1$. Let $k\geq 2$ be an integer and let $T$ be a spanning tree of a connected graph.
The total $k$-excess $te(T,k)$ is the summation of the $k$-excesses of all vertices in $T$, namely, $te(T,k)=\sum_{v\in V(T)}\mbox{max}\{0, d_{T}(v)-k\}.$ One can see that $T$ is a spanning $k$-tree if and only if $te(T,k)=0$. In this paper, we present a tight sufficient condition in terms of the spectral radius for a connected $1$-binding graph to contain a spanning tree with bounded total $k$-excess, which generalizes the result of Fan, Liu and Ao [Linear Algebra Appl. 705 (2025) 1-16] on the existence of a spanning $k$-tree in $1$-binding graphs.

The toughness $\tau(G)=\mathrm{min}\{\frac{|S|}{c(G-S)}: S~\mbox{is a cut set of vertices in}~G\}$ for $G\ncong K_n$. A graph $G$ is called $t$-tough if $\tau(G)\geq t.$ We in this paper also provide a tight sufficient condition based on the spectral radius for a connected $\frac{1}{k-\eta}$-tough graph to contain a spanning tree with bounded total $k$-excess, where $k\geq3$ is an integer and $\eta=\{0,1\}$. It extends the result of Liu, Fan and Shu [Discrete Math. 348 (2025) 114593] on the existence of a spanning $k$-tree in connected $\frac{1}{k-\eta}$-tough graphs.