  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.
