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 Ξ by β bonds connecting the same adjacent
vertices, thereby yielding the lattice Ξββ. This relation is used to
calculate the bond percolation threshold on Ξββ. We show that this
bond inflation leaves the universality class of the percolation transition
invariant on a lattice of dimensionality dβ₯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 Nββ
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 Nββ.Comment: 33 pages latex 3 figure