Branched polymers with prescribed number of cycles: Monte Carlo and exact series studies

Using the newly developed incomplete enumeration method, the number $A_N$(C) and the mean-square radii of gyration 〈$R_N^2$(C)〉 of N-site lattice animals with cyclomatic number C have been studied up to N=70,C=7 on the square and N=30,C=5 on the simple cubic lattices. These are compared with extended exact series data up to N=17,C=4 on the square and N=12,C=4 on the simple cubic lattices. The result is consistent with ${\lambda }_C$=${\lambda }_0$, ${\mathrm{theta}}_C$=${\mathrm{theta}}_0$-C, and ${\nu }_C$=${\nu }_0$, for all C, where ${\lambda }_C$ is the growth parameter, and the exponents ${\mathrm{theta}}_C$ and ${\nu }_C$ characterize the number and the mean-square radii of gyration, respectively, of lattice animals with C cycles.