  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 $\Lambda$ by $\ell$ bonds connecting the same adjacent
vertices, thereby yielding the lattice $\Lambda_\ell$. This relation is used to
calculate the bond percolation threshold on $\Lambda_\ell$. We show that this
bond inflation leaves the universality class of the percolation transition
invariant on a lattice of dimensionality $d \ge 2$ 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 $N \to \infty$
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 $N \to \infty$.
