Two-dimensional lattice embeddings of connected graphs of cyclomatic index two

Abstract

The authors are concerned with the number of figure eights, dumbbells and theta graphs weakly embeddable in a two-dimensional lattice. They show rigorously that, for the square lattice, the dominant limiting behaviour of the numbers of dumbbells and theta graphs is the same as the limiting behaviour of the number of self-avoiding walks and that the total number of figure eights is less than or equal to the number of rooted polygons. Numerical data are presented to suggest that this bound may be the best possible. Estimates of critical exponents for dumbbells and theta graphs are also obtained.