Related papers: Torsor Structures on Spanning Trees
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 $H^{(3)}$ acting on the…
Reflection and braid equations for rank two $q$-tensors are derived from the covariance properties of quantum vectors by using the $R$-matrix formalism.
If g is a quasitriangular Lie bialgebra, one can asks what is the geometrical meaning of its r-matrix. A first answer was given in a paper by Weinstein and Xu, using purely geometrical means: roughly, one has that the formal Poisson group…
This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, which are called irreducible triangulations. The structure has been introduced by Xin He under…
We consider three probability measures on subsets of edges of a given finite graph $G$, namely those which govern, respectively, a uniform forest, a uniform spanning tree, and a uniform connected subgraph. A conjecture concerning the…
A celebrated result of Otter says the number of distinct unlabelled spanning trees in $K_n$ is $\alpha^n$ up to subexponential factors for an absolute constant $\alpha>0$. In this note, we prove that for every $0<\varepsilon<\alpha$, there…
We present the construction of some kind of "convex core" for the product of two actions of a group on $\bbR$-trees. This geometric construction allows to generalize and unify the intersection number of two curves or of two measured…
The classical matrix tree theorem relates the number of spanning trees of a connected graph with the product of the nonzero eigenvalues of its Laplacian matrix. The class of regular matroids generalizes that of graphical matroids, and a…
For their famous algorithm for the disjoint paths problem, Robertson and Seymour proved that there is a function $f$ such that if the tree-width of a graph $G$ with $k$ pairs of terminals is at least $f(k)$, then $G$ contains a…
Any function can be constructed using a hierarchy of simpler functions through compositions. Such a hierarchy can be characterized by a binary rooted tree. Each node of this tree is associated with a function which takes as inputs two…
We study straight-line drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every 3-connected…
In this paper, we study boundary actions of CAT(0) spaces from a point of view of topological dynamics and $C^*$-algebras. First, we investigate the actions of right-angled Coexter groups and right-angled Artin groups with finite defining…
In a series of four papers we determine structures whose existence is dual, in the sense of complementary, to the existence of stars or combs. Here, in the third paper of the series, we present duality theorems for a combination of stars…
For a weighted digraph without loops $V$, the arc weights of which can be obtained from an undirected graph with loops ${\sf P}$ according to the rule $v_{ij}=p_{ij}-p_{ii}$, the properties are studied. An effective algorithm for…
We present analytical results for massless correlators of two vector, tensor, and scalar composite vertices with the Bjorken fractions $x$ and $y$ of order $\alpha_s^2 \beta_0$ of QCD. The structure of these correlators…
Let (R,m) be a local ring and U_R=Spec(R) -{m} be the punctured spectrum of R. Gabber conjectured that if R is a complete intersection of dimension 3, then the abelian group Pic(U_R) is torsion-free. In this note we prove Gabber's statement…
We prove that any planar graph on $n$ vertices has less than $O(5{.}2852^n)$ spanning trees. Under the restriction that the planar graph is 3-connected and contains no triangle and no quadrilateral the number of its spanning trees is less…
To a given nonsingular triangular matrix A with entries from a ring, we associate a weighted bipartite graph G(A) and give a combinatorial description of the inverse of A by employing paths in G(A). Under a certain condition, nonsingular…
Product structure theorems are a collection of recent results that have been used to resolve a number of longstanding open problems on planar graphs and related graph classes. One particularly useful version states that every planar graph…
In this paper, we study the structure of braid graphs in simply-laced Coxeter systems. We prove that every reduced expression has a unique factorization as a product of so-called links, which in turn induces a decomposition of the braid…