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The probability that two sites of a crystal lattice belong to the same cluster (the pair connectedness) is shown to play an important role in percolation theory. Use of the linked-graph method to obtain low-density series expansions leads to the discussion of topological invariants for rooted graphs. These are related to the k-weights which arose in a previous investigation of the mean number of clusters. The mean size of clusters is related to the pair connectedness via sum rules.
J W Essam (Sat,) studied this question.
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