On Vertex Attack Tolerance in Regular Graphs

We have previously introduced vertex attack tolerance (VAT), denoted mathematically as $\tau(G) = \min_{S \subset V} \frac{|S|}{|V-S-C_{max}(V-S)|+1}$ where $C_{max}(V-S)$ is the largest connected component in $V-S$, as an appropriate mathematical measure of resilience in the face of targeted node attacks for arbitrary degree networks. A part of the motivation for VAT was the observation that, whereas conductance, $\Phi(G)$, captures both edge based and node based resilience for regular graphs, conductance fails to capture node based resilience for heterogeneous degree distributions. We had previously demonstrated an upper bound on VAT via conductance for the case of $d$-regular graphs $G$ as follows: $\tau(G) \le d\Phi(G)$ if $\Phi(G) \le \frac{1}{d^2}$ and $\tau(G) \le d^2\Phi(G)$ otherwise. In this work, we provide a new matching lower bound: $\tau(G) \ge \frac{1}{d}\Phi(G)$. The lower and upper bound combined show that $\tau(G) = \Theta(\Phi(G))$ for regular constant degree $d$ and yield spectral bounds as corollaries.