  It has been recently proposed that the natural connectivity can be used to
characterize efficiently the robustness of complex networks. The natural
connectivity quantifies the redundancy of alternative routes in the network by
evaluating the weighted number of closed walks of all lengths and can be seen
as an average eigenvalue obtained from the graph spectrum. In this paper, we
explore both analytically and numerically the natural connectivity of regular
ring lattices and regular random graphs obtained through degree-preserving
random rewirings from regular ring lattices. We reformulate the natural
connectivity of regular ring lattices in terms of generalized Bessel functions
and show that the natural connectivity of regular ring lattices is independent
of network size and increases with monotonically. We also show that random
regular graphs have lower natural connectivity, and are thus less robust, than
regular ring lattices.
