Some Properties of Weight Factors arising in Low-Density Series Expansion for Percolation Models

Abstract

Let F(G) be any additive property of a simple graph such that F(G)=F(G₁)+F(G₂), where G is the series combination of graphs G₁ and G₂. 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)=∑_{G^′⊆ G}(-1)^e-e′F(G^′)η(G^′), where η(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(G₂)W(G₁)+d(G₁)W(G₂), where d(G₁) are d(G₂) the d-weight for graphs G₁ and G₂ 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.