Reciprocal relations for the weight factors arising in the series expansion of backbone percolation functions

The authors consider several properties of the 'backbone' of percolation clusters. The percolation model on a graph G is defined as follows. Each element (vertex or edge) of G is either 'open' or 'closed', the element e being open with probability pe independently of all other elements. The u-v backbone in any configuration is the subgraph consisting of all elements which belong to at least one open, self-avoiding, path from a vertex u to a vertex v. They show that the expected value of any u-v backbone random variable Z, which depends only on the elements of the u-v backbone (e.g. Z may be the number of edges or the length of the shortest path of the backbone) is given by E(Z)= Sigma K contained in P(-1)mod K mod +1ZK Pi e in g(K)Pe where P is the set of all self-avoiding paths from u to v on G, g(K) is the subgraph formed by all elements which belong to at least one self-avoiding path of K and the weightZK is given by ZK= Sigma J contained in K(-1)mod Jmod +1 ZJ where ZJ is the value of Z when the backbone is g(J). Certain 'reciprocal' relations are found to hold between the weights and the backbone variables in some important cases. Using these results, series expansions for the various properties can be obtained.