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) not equal 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. Conjectured by Brouwer and proved by Gu, a theorem asserts that for any d-regular connected graph G, the toughness t(G) is always at least lambda/d - 1, where lambda 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.