Stability Theorems for Graph Vulnerability Parameters

Spectral Conditions, Degree Sequences, and Graphical Properties

“…Next, we consider stability results for the same graph properties, which are obtained by Yatauro in [7].…”

“…Mathematics 2023, 11, 4264. https://doi.org/10.3390/math11204264 https://www.mdpi.com/journal/mathematics where m(G − X) is the order of the largest component of G − X and ω(G − X) is the number of components of G − X. As in [7], a graph G is k-integral if I(G) ≥ k, k-tough if τ(G) ≥ k, k-tenacious if T(G) ≥ k, and k-binding if bind (G) ≥ k.…”

“…Moreover, Cunningham [15] has shown that the binding number bind (G) is tractable. In particular, Yatauro [7] recently obtained the stability theorems for the properties of G being k-integral, k-tough, k-tenacious, or k-binding. In general, we hardly determine integrity, tenacity, binding number, or toughness by the definitions of these parameters, i.e., the equations in (1).…”

“…After that, Feng et al [33] considered some graph properties with the spectral radius, including k-connected, k-edge-connected, k-Hamiltonian, k-edge-Hamiltonian, β-deficient, and k-path-coverable. Recently, the degree sequence is also used in a graph to determine if it is k-integral [34], k-tenacious [11], k-binding [35], or k-tough [36]. In this paper, according to the methods in [33], we shall utilize the degree sequence and the closure concepts to get several sufficient conditions of graphs with certain properties, including k-integral, k-tenacious, k-binding, and k-tough.…”