Patterns in molecular topology and complexity for long-chain branching are
quantitatively described. The Wiener number, the topological complexity ind
ex, and a new index of 3-starness are used to quantify polymer structure. G
eneral formulas for these indices were derived for the cases of 3-arm star,
H-shaped, and B-arm comb polymers. The factors affecting complexity in mon
odisperse polymer systems are ranked as follows: number of arms much greate
r than arm length > arm central position;: arm clustering > total molecular
weight backbone molecular weight. Topological indices change rapidly and t
hen plateau as the molecular weight of branches on a polyolefin backbone in
creases from 0 to 5 kD. Complexity calculations relate 2-arm or 3-arm comb
structures to the corresponding 3-arm stars of equivalent complexity but mu
ch higher molecular weight. In a subsequent paper, we report the applicatio
n of topological analysis for developing structure/property relationships f
or monodisperse polymers. While the focus of the present work is on the des
cription of monodisperse, well-defined architectures, the methods may be ex
tended to the description of polydisperse systems.