A complete diagrammatic expansion is developed for the Domb-Joyce model of an
N
-step chain, with an interaction
w
which varies between 0 and 1. Simple rules are given for obtaining the diagrams. The correspondence between these diagrams and appropriate generating functions permits computation of the coefficients of the series
α
2
N
(
w
) = 1 +
k
1
w
+
k
2
w
2
+ . . ., where
α
2
N
(
w
) is the expansion factor of the mean square end-to-end length of the chain. The dominant term in
N
of each of the first three
k
r
is shown to be identical for the three cubic lattices and for the Gaussian continuum model, with the exception of a scale factor
h
0
. Retention of only this dominant term yields a ‘two-parameter’ expansion equivalent to that of Zimm (1946), Fixman (1955) and others. Diagrams are classed either as ladder or as non-ladder graphs. The ladder graph contributions are summed by using functional relations of Domb & Joyce (1972). The non-ladder contributions for the first three coefficients are computed individually, thereby yielding results for
k
1
,
k
2
and
k
3
in terms of the ‘universal’ parameter
z
=
h
0
N
1/2
w
. The terms
k
1
and
k
2
agree with previous computations for the Gaussian model but
k
3
differs slightly.
Pair difference cordiality of some graphs derived from ladder graph
