The number of independent sets is equivalent to the partition function of the hard-core lattice gas model with nearest-neighbor exclusion and unit activity. We study the number of independent sets $m_{d,b}(n)$ 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 for $d=2$. The upper and lower bounds for the asymptotic growth constant, defined as $z_{SG_{d,b}}=lim_{v to infty} ln m_{d,b}(n)/v$ where $v$ is the number of vertices, on these Sierpinski gaskets are derived in terms of the results at a certain stage. The numerical values of these $z_{SG_{d,b}}$ are evaluated with more than a hundred significant figures accurate. We also conjecture the upper and lower bounds for the asymptotic growth constant $z_{SG_{d,2}}$ with general $d$.
The energy of a simple graph $G$ arising in chemical physics, denoted by $mathcal E(G)$, is defined as the sum of the absolute values of eigenvalues of $G$. We consider the asymptotic energy per vertex (say asymptotic energy) for lattice systems. In
general for a type of lattice in statistical physics, to compute the asymptotic energy with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions are different tasks with different hardness. In this paper, we show that if ${G_n}$ is a sequence of finite simple graphs with bounded average degree and ${G_n}$ a sequence of spanning subgraphs of ${G_n}$ such that almost all vertices of $G_n$ and $G_n$ have the same degrees, then $G_n$ and $G_n$ have the same asymptotic energy. Thus, for each type of lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions, we have the same asymptotic energy. As applications, we obtain the asymptotic formulae of energies per vertex of the triangular, $3^3.4^2$, and hexagonal lattices with toroidal, cylindrical, Mobius-band, Klein-bottle, and free boundary conditions simultaneously.
