489 research outputs found
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
Enumeration of noncrossing trees on a circle
AbstractWe consider several enumerative problems concerning labelled trees whose vertices lie on a circle and whose edges are rectilinear and do not cross
The electrical response matrix of a regular 2n-gon
Consider a unit-resistive plate in the shape of a regular polygon with 2n
sides, in which even-numbered sides are wired to electrodes and odd-numbered
sides are insulated. The response matrix, or Dirichlet-to-Neumann map, allows
one to compute the currents flowing through the electrodes when they are held
at specified voltages. We show that the entries of the response matrix of the
regular 2n-gon are given by the differences of cotangents of evenly spaced
angles, and we describe some connections with the limiting distributions of
certain random spanning forests.Comment: 10 pages, 4 figures; v2 adds more background informatio
Random enriched trees with applications to random graphs
We establish limit theorems that describe the asymptotic local and global
geometric behaviour of random enriched trees considered up to symmetry. We
apply these general results to random unlabelled weighted rooted graphs and
uniform random unlabelled -trees that are rooted at a -clique of
distinguishable vertices. For both models we establish a Gromov--Hausdorff
scaling limit, a Benjamini--Schramm limit, and a local weak limit that
describes the asymptotic shape near the fixed root
Logarithmic two-point correlators in the Abelian sandpile model
We present the detailed calculations of the asymptotics of two-site
correlation functions for height variables in the two-dimensional Abelian
sandpile model. By using combinatorial methods for the enumeration of spanning
trees, we extend the well-known result for the correlation of minimal heights to for
height values . These results confirm the dominant logarithmic
behaviour for
large , predicted by logarithmic conformal field theory based on field
identifications obtained previously. We obtain, from our lattice calculations,
the explicit values for the coefficients and (the latter are new).Comment: 28 page
Spanning Trees of Lattices Embedded on the Klein Bottle
The problem of enumerating spanning trees in lattices with Klein bottle boundary condition is considered here. The exact closed-form expressions of the numbers of spanning trees for 4.8.8 lattice, hexagonal lattice, and 33·42 lattice on the Klein bottle are presented
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