We study the number of connected spanning subgraphs f d,b (n) on the generalized Sierpinski gasket SG d,b (n) at stage n with dimension d equal to two, three and four for b = 2, and layer b equal to three and four for d = 2. The upper and lower bounds for the asymptotic growth constant, defined as z SG d,b = lim v→∞ ln f d,b (n)/v where v is the number of vertices, on SG 2,b (n) with b = 2, 3, 4 are derived in terms of the results at a certain stage. The numerical values of z SG d,bare obtained.

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We present exact calculations of the partition function of the q-state Potts model for general q and temperature on strips of the square lattice of width L y = 3 vertices and arbitrary length L x with periodic longitudinal boundary conditions, of the following types: (i) (F BC y , P BC x ) = cyclic, (ii) (F BC y , T P BC x ) = Möbius, (iii) (P BC y , P BC x ) = toroidal, and (iv) (P BC y , T P BC x ) = Klein bottle, where F BC and (T )P BC refer to free and (twisted) periodic boundary conditions. Results for the L y = 2 torus and Klein bottle strips are also included. In the infinite-length limit the thermodynamic properties are discussed and some general results are given for low-temperature behavior on strips of arbitrarily great width. We determine the submanifold in the C 2 space of q and temperature where the free energy is singular for these strips. Our calculations are also used to compute certain quantities of graph-theoretic interest. *