Some Properties of Weight Factors arising in Low-Density Series Expansion for Percolation Models - Bhatti, Faqir M; Abu, Nur Azman (2002)
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) = Sigma(G'subset of or equal toG)(-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.