Expansion Expressions for si and

As noted in Section 2, the fundamental bond function in cluster expansion theory is the Mayer / function defined in Eq. (8). Both si and g can be expressed very simply in terms of graphical series containing / bonds. 

Vsiifi,pi. py) = sum of all topologically different irreducible graphs that have no root points, two or more field points, and at most one / bond between each pair of points (16) 

The first few diagrams in each series are shown in Fig. 5. These expressions are the starting point for all the applications to be discussed below. The derivation of these results from the fundamental statistical-mechanical expressions for si and g is an interesting and challenging exercise in graph theory. 

Expressions for h and y can now easily be obtained. The expression for g contains one diagram with no bond. Its value is unity. All the other diagrams in g have at least one bond. 

X2) = sum of all graphs in Eq. (17) for g that have at least one bond. The expression for y can be obtained by noting from Eq. (15) that 

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