Related papers: Invariant spanning trees for quadratic rational ma…
Thurston maps are branched self-coverings of the sphere whose critical points have finite forward orbits. We give combinatorial and algebraic characterizations of Thurston maps that are isotopic to expanding maps as "Levy-free" maps and as…
This is the announcement, and the long summary, of a series of articles on the algorithmic study of Thurston maps. We describe branched coverings of the sphere in terms of group-theoretical objects called bisets, and develop a theory of…
In this paper, we introduce a magneto-spectral invariant for finite graphs. This invariant vanishes on trees and is maximized by complete graphs. We compute this invariant for cycles, complete graphs, wheel graphs, hypercubes, complete…
We consider "Thurston maps": branched self-coverings of the sphere with ultimately periodic critical points, and prove that the Thurston equivalence problem between them (continuous deformation of maps along with their critical orbits) is…
We consider Thurston maps, i.e., branched covering maps $f\colon S^2\to S^2$ that are postcritically finite. In addition, we assume that $f$ is expanding in a suitable sense. It is shown that each sufficiently high iterate $F=f^n$ of $f$ is…
Among Thurston maps (orientation-preserving, postcritically finite branched coverings of the 2-sphere to itself), those that arise as subdivision maps of a finite subdivision rule form a special family. For such maps, we investigate…
We extend the concept of a Hubbard tree, well established and useful in the theory of polynomial dynamics, to the dynamics of transcendental entire functions. We show that Hubbard trees in the strict traditional sense, as invariant compact…
In earlier work, we constructed invariants of irreducible representations of the Kauffman skein algebra of a surface. We introduce here an inverse construction, which to a set of possible invariants associates an irreducible representation…
A study of real quadratic maps with real critical points, emphasizing the effective construction of critically finite maps with specified combinatorics. We discuss the behavior of the Thurston algorithm in obstructed cases, and in one…
The spanning tree heuristic is a commonly adopted procedure in network inference and estimation. It allows one to generalize an inference method developed for trees, which is usually based on a statistically rigorous approach, to a…
In this paper we give an exact analytical expression for the number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. This number is an important graph invariant related to different topological…
We use the theory of self-similar groups to enumerate all combinatorial classes of non-exceptional quadratic Thurston maps with fewer than five postcritical points. The enumeration relies on our computation that the corresponding maps on…
A Thurston map is a branched covering map $f\colon S^2\to S^2$ that is postcritically finite. Mating of polynomials, introduced by Douady and Hubbard, is a method to geometrically combine the Julia sets of two polynomials (and their…
We develop a general theory of "bisets": sets with two commuting group actions. They naturally encode topological correspondences. Just as van Kampen's theorem decomposes into a graph of groups the fundamental group of a space given with a…
We consider each of the three classes of representations of cyclic groups that arise in the study of rational sphere maps. We study the possible number of terms for invariant polynomials with non-negative coefficients that are constant on…
Variable trees are a new method for the exploration of discrete multivariate data. They display nested subsets and corresponding frequencies and percentages. Manual calculation of these quantities can be laborious, especially when there are…
Graph invariants are a useful tool in graph theory. Not only do they encode useful information about the graphs to which they are associated, but complete invariants can be used to distinguish between non-isomorphic graphs. Polynomial…
We give a simple algorithm that determines whether a given post-critically finite topological polynomial is Thurston equivalent to a polynomial. If it is, the algorithm produces the Hubbard tree; otherwise, the algorithm produces the…
The quadratic minimum spanning tree problem and its variations such as the quadratic bottleneck spanning tree problem, the minimum spanning tree problem with conflict pair constraints, and the bottleneck spanning tree problem with conflict…
Symplectic invariants introduced in math-ph/0702045 can be computed for an arbitrary spectral curve. For some examples of spectral curves, those invariants can solve loop equations of matrix integrals, and many problems of enumerative…