44 research outputs found
Spanning forests on the Sierpinski gasket
We present the numbers of spanning forests on the Sierpinski gasket
at stage with dimension equal to two, three and four, and determine the
asymptotic behaviors. The corresponding results on the generalized Sierpinski
gasket with and are obtained. We also derive the
upper bounds of the asymptotic growth constants for both and .Comment: 31 pages, 9 figures, 7 table
Dimer-monomer model on the Sierpinski gasket
We present the numbers of dimer-monomers on the Sierpinski gasket
at stage with dimension equal to two, three and four, and determine the
asymptotic behaviors. The corresponding results on the generalized Sierpinski
gasket with and are obtained.Comment: 30 pages, 10 figures, 10 table
Tutte polynomial of pseudofractal scale-free web
The Tutte polynomial of a graph is a 2-variable polynomial which is quite
important in both combinatorics and statistical physics. It contains various
numerical invariants and polynomial invariants, such as the number of spanning
trees, the number of spanning forests, the number of acyclic orientations, the
reliability polynomial, chromatic polynomial and flow polynomial. In this
paper, we study and gain recursive formulas for the Tutte polynomial of
pseudofractal scale-free web (PSW) which implies logarithmic complexity
algorithm is obtained to calculate the Tutte polynomial of PSW although it is
NP-hard for general graph. We also obtain the rigorous solution for the the
number of spanning trees of PSW by solving the recurrence relations derived
from Tutte polynomial, which give an alternative approach for explicitly
determining the number of spanning trees of PSW. Further more, we analysis the
all-terminal reliability of PSW and compare the results with that of Sierpinski
gasket which has the same number of nodes and edges with PSW. In contrast with
the well-known conclusion that scale-free networks are more robust against
removal of nodes than homogeneous networks (e.g., exponential networks and
regular networks). Our results show that Sierpinski gasket (which is a regular
network) are more robust against random edge failures than PSW (which is a
scale-free network). Whether it is true for any regular networks and scale-free
networks, is still a unresolved problem.Comment: 19pages,7figures. arXiv admin note: text overlap with arXiv:1006.533
Number of connected spanning subgraphs on the Sierpinski gasket
We study the number of connected spanning subgraphs on the
generalized Sierpinski gasket at stage with dimension
equal to two, three and four for , and layer equal to three and four
for . The upper and lower bounds for the asymptotic growth constant,
defined as where is the
number of vertices, on with are derived in terms of the
results at a certain stage. The numerical values of are
obtained.Comment: 25 pages, 8 figures, 9 table
Zeros of the Potts Model Partition Function on Sierpinski Graphs
We calculate zeros of the -state Potts model partition function on
'th-iterate Sierpinski graphs, , in the variable and in a
temperature-like variable, . We infer some asymptotic properties of the loci
of zeros in the limit and relate these to thermodynamic
properties of the -state Potts ferromagnet and antiferromagnet on the
Sierpinski gasket fractal, .Comment: 6 pages, 8 figure
Weighted spanning trees on some self-similar graphs
We compute the complexity of two infinite families of finite graphs: the
Sierpi\'{n}ski graphs, which are finite approximations of the well-known
Sierpi\'nsky gasket, and the Schreier graphs of the Hanoi Towers group
acting on the rooted ternary tree. For both of them, we study the
weighted generating functions of the spanning trees, associated with several
natural labellings of the edge sets.Comment: 21 page