Related papers: Rotor-routing and spanning trees on planar graphs

The aim of this note is to extend the result of Angel and Holroyd concerning the transience and the recurrence of transfinite rotor-router walks, for random initial configuration of rotors on homogeneous trees. We address the same question…

Tree-width and its linear variant path-width play a central role for the graph minor relation. In particular, Robertson and Seymour (1983) proved that for every tree~$T$, the class of graphs that do not contain $T$ as a minor has bounded…

The rotor-router model is a deterministic analogue of random walk. It can be used to define a deterministic growth model analogous to internal DLA. We show that the set of occupied sites for this model on an infinite regular tree is a…

We give a rigorous and self-contained survey of the abelian sandpile model and rotor-router model on finite directed graphs, highlighting the connections between them. We present several intriguing open problems.

This paper provides some new guidance in the construction of region graphs for Generalized Belief Propagation (GBP). We connect the problem of choosing the outer regions of a LoopStructured Region Graph (SRG) to that of finding a…

For a directed graph G on vertices {0,1,...,n}, a G-parking function is an n-tuple (b_1,...,b_n) of non-negative integers such that, for every non-empty subset U of {1,...,n}, there exists a vertex j in U for which there are more than b_j…

We study a notion of deformation for simplicial trees with group actions (G-trees). Here G is a fixed, arbitrary group. Two G-trees are related by a deformation if there is a finite sequence of collapse and expansion moves joining them. We…

We consider the symplectic action of a finite group G on a K3 surface. The Picard group of the K3 surface has a primitive sublattice determined by G. We show how to compute the rank and discriminant of this sublattice. We then describe…

It is shown that for any action of a finitely presented group $G$ on an $\R$-tree, there is a decomposition of $G$ as the fundamental group of a graph of groups related to this action. If the action of $G$ on $T$ is non-trivial, i.e. there…

In [5], Holroyd, Levine, M\'esz\'aros, Peres, Propp and Wilson characterize recurrent chip-and-rotor configurations for strongly connected digraphs. However, the number of steps needed to recur, and the number of orbits is left open for…

The rotor-router model on a graph describes a discrete-time walk accompanied by the deterministic evolution of configurations of rotors randomly placed on vertices of the graph. We prove the following property: if at some moment of time,…

Building on recent work of Robertson and Steger, we associate a C*-algebra to a combinatorial object which may be thought of as a higher rank graph. This C*-algebra is shown to be isomorphic to that of the associated path groupoid.…

We show that on any Riemann surface S of genus g>1 any nonsingular even spin bundle defines e-foloation of S. When a surface is hyperelliptic then all leaves of this foliation are finite and almost all of them consists of 2g+2 points.…

For a graph G, we construct two algebras, whose dimensions are both equal to the number of spanning trees of G. One of these algebras is the quotient of the polynomial ring modulo certain monomial ideal, while the other is the quotient of…

Combining the study of the simple random walk on graphs, generating functions (especially Green functions), complex dynamics and general complex analysis we introduce a new method of spectral analysis on self-similar graphs. We give an…

We present a link-by-link rule-based method for constructing all members of the ensemble of spanning trees for any recursively generated, finitely articulated graph, such as the DGM net. The recursions allow for many large-scale properties…

We give a deterministic algorithm to construct a graph with no loops (a tree or a forest) whose vertices are the points of a d-dimensional stationary Poisson process S, subset of R^d. The algorithm is independent of the origin of…

Let G be a reductive group over a commutative ring R. We say that G has isotropic rank >=n, if every normal semisimple reductive R-subgroup of G contains (G_m)^n. We prove that if G has isotropic rank >=1 and R is a regular domain…

We introduce a class of spaces, called real cubings, and study the stucture of groups acting nicely on these spaces. Just as cubings are a natural generalisation of simplicial trees, real cubings can be regarded as a natural generalisation…

An independent set in a graph is a set of pairwise non-adjacent vertices. Let $\alpha(G)$ denote the cardinality of a maximum independent set in the graph $G = (V, E)$. Gutman and Harary defined the independence polynomial of $G$ \[ I(G;x)…