We have previously introduced vertex attack tolerance (VAT), denoted mathematically as τ(G) = minS⊂V |S| |V−S−Cmax(V−S)|+1 where Cmax(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, Φ(G), captu...

A graph is one-regular if its automorphism group acts regularly on the arc set. In this paper, we construct a new infinite family of one-regular Cayley graphs of any prescribed valency. In fact, for any two positive integers , k 2 except for ( , k) ∈ {(2,3), (2,4)}, the Cayley graph Cay(Dn,S) on dihedral groups Dn = 〈a, b | an = b2 = (ab)2 = 1〉 with S = {a1+ +···+ t b | 0 t k − 1} and n = ∑k−1 ...