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Register a new userDimer coverings on the Sierpinski gasket with possible vacancies on the outmost vertices
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We present the number of dimers $N_d(n)$ on the Sierpinski gasket $SG_d(n)$ at stage $n$ with dimension $d$ equal to two, three, four or five, where one of the outmost vertices is not covered when the number of vertices $v(n)$ is an odd number. The entropy of absorption of diatomic molecules per site, defined as $S_{SG_d}=lim_{n to infty} ln N_d(n)/v(n)$, is calculated to be $ln(2)/3$ exactly for $SG_2(n)$. The numbers of dimers on the generalized Sierpinski gasket $SG_{d,b}(n)$ with $d=2$ and $b=3,4,5$ are also obtained exactly. Their entropies are equal to $ln(6)/7$, $ln(28)/12$, $ln(200)/18$, respectively. The upper and lower bounds for the entropy are derived in terms of the results at a certain stage for $SG_d(n)$ with $d=3,4,5$. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of $S_{SG_d}$ with $d=3,4,5$ can be evaluated with more than a hundred significant figures accurate.
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We derive exactly the number of Hamiltonian paths H(n) on the two dimensional Sierpinski gasket SG(n) at stage $n$, whose asymptotic behavior is given by $frac{sqrt{3}(2sqrt{3})^{3^{n-1}}}{3} times (frac{5^2 times 7^2 times 17^2}{2^{12} times 3^5 times 13})(16)^n$. We also obtain the number of Hamiltonian paths with one end at a certain outmost vertex of SG(n), with asymptotic behavior $frac {sqrt{3}(2sqrt{3})^{3^{n-1}}}{3} times (frac {7 times 17}{2^4 times 3^3})4^n$. The distribution of Hamiltonian paths on SG(n) with one end at a certain outmost vertex and the other end at an arbitrary vertex of SG(n) is investigated. We rigorously prove that the exponent for the mean $ell$ displacement between the two end vertices of such Hamiltonian paths on SG(n) is $ell log 2 / log 3$ for $ell>0$.
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 $d$.
We present the numbers of ice model and eight-vertex model configurations (with Boltzmann factors equal to one), I(n) and E(n) respectively, on the two-dimensional Sierpinski gasket SG(n) at stage $n$. For the eight-vertex model, the number of configurations is $E(n)=2^{3(3^n+1)/2}$ and the entropy per site, defined as $lim_{v to infty} ln E(n)/v$ where $v$ is the number of vertices on SG(n), is exactly equal to $ln 2$. For the ice model, the upper and lower bounds for the entropy per site $lim_{v to infty} ln I(n)/v$ are derived in terms of the results at a certain stage. As the difference between these bounds converges quickly to zero as the calculated stage increases, the numerical value of the entropy can be evaluated with more than a hundred significant figures accurate. The corresponding result of ice model on the generalized two-dimensional Sierpinski gasket SG_b(n) with $b=3$ is also obtained. For the generalized vertex model on SG_3(n), the number of configurations is $2^{(8 times 6^n +7)/5}$ and the entropy per site is equal to $frac87 ln 2$. The general upper and lower bounds for the entropy per site for arbitrary $b$ are conjectured.
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 asymptotic growth constants for $SG_{2,b}$ and $d$-dimensional Sierpinski gasket $SG_d$.
As a basic dynamic feature on complex networks, the property of random walk has received a lot of attention in recent years. In this paper, we first studied the analytical expression of the mean global first passage time (MGFPT) on the 3-dimensional 3-level Sierpinski gasket network. Based on the self-similar structure of the network, the correlation between the MGFPT and the average trapping time (ATT) is found, and then the analytical expression of the ATT is obtained. Finally, by establishing a joint network model, we further give the standard process of solving the analytical expression of the ATT when there is a set of trap nodes in the network. By illustrating examples and numerical simulations, it can be proved that when the trap node sets are different, the ATT will be quite different, but the the super-linear relationship with the number of iterations will not be changed.
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