Binding Number, k-Factor and Spectral Radius of Graphs

The binding number $b(G)$ of a graph G is the minimum value of |NG(X)|/|X| taken over all non-empty subsets X of $V(G)$ such that NG(X)≠ V(G). The association between the binding number and toughness is intricately interconnected, as both metrics function as pivotal indicators for quantifying the vulnerability of a graph. The Brouwer-Gu Theorem asserts that for any d-regular connected graph G, the toughness $t(G)$ always at least d/λ-1, where λ denotes the second largest absolute eigenvalue of the adjacency matrix. Inspired by the work of Brouwer and Gu, in this paper, we investigate $b(G)$ from spectral perspectives, and provide tight sufficient conditions in terms of the spectral radius of a graph G to guarantee b(G)≥ r. The study of the existence of k-factors in graphs is a classic problem in graph theory. Katerinis and Woodall state that every graph with order n≥ 4k-6 satisfying b(G)≥ 2 contains a k-factor where k≥ 2. This leaves the following question: which $1$-binding graphs have a k-factor? In this paper, we also provide the spectral radius conditions of $1$-binding graphs to contain a perfect matching and a $2$-factor, respectively.