Spanning k-trees, odd [1,b]-factors and spectral radius in binding graphs

TL;DR

This paper establishes spectral radius-based conditions ensuring the existence of odd [1,b]-factors and spanning k-trees in binding graphs, generalizing and improving previous spectral graph theory results.

Contribution

It provides new tight spectral radius conditions for binding graphs to contain odd [1,b]-factors and spanning k-trees, extending prior work in spectral graph theory.

Findings

Spectral radius conditions guarantee odd [1,b]-factors in 1/b-binding graphs.

Spectral radius bounds ensure spanning k-trees in 1/(k-2)-binding graphs.

Results improve and generalize previous spectral conditions in the literature.

Abstract

The binding number of a graph $G$, written as $\mbox{bind}(G)$, is defined by $$ \mbox{bind}(G)=\min\left\{\frac{|N_G(X)|}{|X|}:\emptyset\neq X\subseteq V(G),N_G(X)\neq V(G)\right\}. $$ A graph $G$ is called $r$-binding if $\mbox{bind}(G)\geq r$. An odd $[1,b]$-factor of a graph $G$ is a spanning subgraph $F$ with $d_F(v)\in\{1,3,\ldots,b\}$ for all $v\in V(G)$, where $b\geq1$ is an odd integer. A spanning $k$-tree of a connected graph $G$ is a spanning tree $T$ with $d_T(v)\leq k$ for every $v\in V(G)$. In this paper, we first show a tight sufficient condition with respect to the adjacency spectral radius for connected $\frac{1}{b}$-binding graphs to have odd $[1,b]$-factors, which generalizes Fan and Lin's previous result [D. Fan, H. Lin, Binding number, $k$-factor and spectral radius of graphs, Electron. J. Combin. 31(1) (2024) \#P1.30] and partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16]. Then we put forward a tight sufficient condition via the adjacency spectral radius for connected $\frac{1}{k-2}$-binding graphs to have spanning $k$-trees, which partly improves Fan, Liu and Ao's previous result [A. Fan, R. Liu, G. Ao, Spectral radius, odd $[1,b]$-factor and spanning $k$-tree of 1-binding graphs, Linear Algebra Appl. 705 (2025) 1--16].