Related papers: Number of connected spanning subgraphs on the Sier…

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…

We present the numbers of spanning forests on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski…

We study the number of acyclic orientations on the generalized two-dimensional Sierpinski gasket $SG_{2,b}(n)$ at stage $n$ with $b$ equal to two and three, and determine the asymptotic behaviors. We also derive upper bounds for the…

We obtain the numbers of spanning trees on the Sierpinski gasket $SG_d(n)$ with dimension $d$ equal to two, three and four. The general expression for the number of spanning trees on $SG_d(n)$ with arbitrary $d$ is conjectured. The numbers…

We present the numbers of dimer-monomers on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three and four, and determine the asymptotic behaviors. The corresponding results on the generalized Sierpinski gasket…

Consider spanning trees on the two-dimensional Sierpinski gasket SG(n) where stage $n$ is a non-negative integer. For any given vertex $x$ of SG(n), we derive rigorously the probability distribution of the degree $j \in \{1,2,3,4\}$ at the…

Let $F(G)$ be the number of spanning forests in a graph $G$ and $\mathcal{C}(n,d)$ be the set of all connected $d$-regular simple graphs of order $n$. Define $\widehat{f}_{d}=\liminf_{n\rightarrow \infty}\{F(G)^{1/n}:G\in…

In this paper it is proved that there are constants 0< c_2< c_1 such that an asymptotic formula can be given for the the number of (labeled) n-vertex graphs of diameter d whenever n tends to infinity and 2 < d < n - c_1 (log n). A typical…

Using the method of spectral decimation and a modified version of Kirchhoffs Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely…

For $n$-vertex, $d$-dimensional lattices $\Lambda$ with $d \ge 2$, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present an exact closed-form result for the…

Under certain continuity conditions, we estimate upper and lower box dimension of graph of a function defined on the Sierpinski gasket. We also give an upper bound for Hausdorff dimension and box dimension of graph of function having finite…

We calculate exponential growth constants $\phi$ and $\sigma$ describing the asymptotic behavior of spanning forests and connected spanning subgraphs on strip graphs, with arbitrarily great length, of several two-dimensional lattices,…

Using the method of spectral decimation and a modified version of Kirchhoff's Matrix-Tree Theorem, a closed form solution to the number of spanning trees on approximating graphs to a fully symmetric self-similar structure on a finitely…

For a two-dimensional lattice $\Lambda$ with $n$ vertices, the number of spanning trees $N_{ST}(\Lambda)$ grows asymptotically as $\exp(n z_\Lambda)$ in the thermodynamic limit. We present exact integral expression and numerical value for…

Let $F(G)$ be the number of forests of a graph $G$. Similarly let $C(G)$ be the number of connected spanning subgraphs of a connected graph $G$. We bound $F(G)$ and $C(G)$ for regular graphs and for graphs with fixed average degree. Among…

We consider the doubly infinite Sierpinski gasket graph $SG_0$, rescale it by factor $2^{-n}$, and on the rescaled graphs $SG_n=2^{-n}SG_0$, for every $n\in \mathbb{N}$, we investigate the limit shape of three aggregation models with…

We study the extension problem on the Sierpinski Gasket ($SG$). In the first part we consider minimizing the functional $\mathcal{E}_{\lambda}(f) = \mathcal{E}(f,f) + \lambda \int f^2 d \mu$ with prescribed values at a finite set of points…

We present a detailed study of the evolution of the number of connected components in sub-critical multiplicative random graph processes. We consider a model where edges appear independently after an exponential time at rate equal to the…

For a real c \geq 1 and an integer n, let f(n,c) denote the maximum integer f so that every graph on n vertices contains an induced subgraph on at least f vertices in which the maximum degree is at most c times the minimum degree. Thus, in…

For each natural number $n$ we determine, both asymptotically and exactly, the maximum number of edges an induced subgraph of order $n$ of the $d$-dimension a grid graph ${\ints}^d$ can have. The asymptotic bound is obtained by using a…