On the connectivity index for lattices of nonintegral dimensionality

Abstract

We define the connectivity indexc for an infinite graph by the requirement that to disconnect a subset of at least V points from the rest of the graph requires the deletion of a minimum of S(V) bonds where S(V) ∼ V ^(c-1)/c for large V. For ad-dimensional hypercubical lattice with d integral, c=d. We construct explicit examples of lattices with nonintegral connectivity index c, 1< c< ∞. It is argued that the connectivity index is an important parameter determining the critical behaviour of Hamiltonians on these lattices.