  We derive exact relations between the Potts model partition function, or
equivalently the Tutte polynomial, for a network (graph) $G$ and a network
obtained from $G$ by (i) by replacing each edge (i.e., bond) of $G$ by two or
more edges joining the same vertices, and (ii) by inserting one or more
degree-2 vertices on edges of $G$. These processes are called edge and vertex
inflation, respectively. The physical effects of these edge and vertex
inflations are discussed. We also present exact calculations of these
polynomials for families of networks obtained via the operation (ii) on a
subset of the bonds of the network. Applications of these results include
calculations of some network reliability polynomials. In addition, we evaluate
our results to calculate various quantities of structural interest such as
numbers of spanning trees, etc., and to determine their asymptotic behavior for
large networks.
