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Some properties of weight factors arising in low-density series expansion for percolation models. (English) Zbl 1267.82063

Summary: Let \(F(G)\) be any additive property of a simple graph such that \(F(G)=F(G_1)+F(G_2)\), where \(G\) is the series combination of graphs \(G_1\) and \(G_2\). The weight factor \(W(G)\) which is based on \(F(G)\) arises in the low-density series expansion techniques for percolation models as \(W(G)=\sum_{G'\subseteq G}(-1)^{e-e'}F(G')\eta(G')\), where \(\eta(G')\) is the indicator that \(G'\) cover-able sub-graph or without dangling ends. The purpose of this paper is to prove the weight factor formula for additive property of \(F\) as \(W(G)=d(G_2)W(G_1)+d(G_1)W(G_2)\), where \(d(G_1)\) are \(d(G_2)\) the d-weight for graphs \(G_1\) and \(G_2\), respectively. This result will be more simplified in the case of Directed Percolation Models using Mobius function property. A new few formulas for the resistive weight factors are also derived for a graph, which is parallel combination of \(n\) edges.

MSC:

82B43 Percolation
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