Branched polymers on the Given-Mandelbrot family of fractals

We study the average number A (n) per site of the number of different configurations of a branched polymer of n bonds on the Given-Mandelbrot family of fractals using exact real-space renormalization. Different members of the family are characterized by an integer parameter b , 2 < or = b < or = infinity . The fractal dimension varies from log(2) 3 to 2 as b is varied from 2 to infinity. We find that for all b > or = 3 , A (n) varies as lambda(n) exp (b n(psi)) where lambda and b are some constants, and 0 < psi < 1 . We determine the exponent psi, and the size exponent nu (average diameter of polymer varies as n(nu) ), exactly for all b , 3 < or = b < o r= infinity . This generalizes the earlier results of Knezevic and Vannimenus for b = 3 [Phys. Rev B 35, 4988 (1987)].