Some Exact Results on Bond Percolation

We present some exact results on bond percolation. We derive a relation that specifies the consequences for bond percolation quantities of replacing each bond of a lattice ?? by ? bonds connecting the same adjacent vertices, thereby yielding the lattice ?? _? . This relation is used to calculate the bond percolation threshold on ?? _? . We show that this bond inflation leaves the universality class of the percolation transition invariant on a lattice of dimensionality d??2 but changes it on a one-dimensional lattice and quasi-one-dimensional infinite-length strips. We also present analytic expressions for the average cluster number per vertex and correlation length for the bond percolation problem on the N???? limits of several families of N-vertex graphs. Finally, we explore the effect of bond vacancies on families of graphs with the property of bounded diameter as N????.