Related papers: On canonical sandpile actions of embedded graphs

Let $C_{k_1}, \ldots, C_{k_n}$ be cycles with $k_i\geq 2$ vertices ($1\le i\le n$). By attaching these $n$ cycles together in a linear order, we obtain a graph called a polygon chain. By attaching these $n$ cycles together in a cyclic…

An important conjecture in percolation theory is that almost surely no infinite cluster exists in critical percolation on any transitive graph for which the critical probability is less than 1. Earlier work has established this for the…

Given a finite connected graph ${\Gamma}$ and a group $G$ acting transitively on the vertices of ${\Gamma}$, we prove that the number of vertices of ${\Gamma}$ and the cardinality of $G$ are bounded above by a function depending only on the…

We show how to construct discrete-time quantum walks on directed, Eulerian graphs. These graphs have tails on which the particle making the walk propagates freely, and this makes it possible to analyze the walks in terms of scattering…

For a smooth (locally trivial) principal bundle in Ehresmann's sense, the relation between the commuting vertical and horizontal actions of the structural Lie group and the structural Lie groupoid (isomorphisms between vertical fibers) is…

We show that any pivotal Hopf monoid $H$ in a symmetric monoidal category $\mathcal{C}$ gives rise to actions of mapping class groups of oriented surfaces of genus $g \geq 1$ with $n \geq 1$ boundary components. These mapping class group…

We prove that a group $\Gamma$ admits a discrete topological (equivalently, smooth) action on some simply-connected 3-manifold if and only if $\Gamma$ has a Cayley complex embeddable -- with certain natural restrictions -- in one of the…

Given a permutation group $G$, the derangement graph of $G$ is the Cayley graph with connection set the derangements of $G$. The group $G$ is said to be innately transitive if $G$ has a transitive minimal normal subgroup. Clearly, every…

We investigate the configuration where a group of finite Morley rank acts definably and generically $m$-transitively on an elementary abelian $p$-group of Morley rank $n$, where $p$ is an odd prime, and $m\geqslant n$. We conclude that…

A maximal minor $M$ of the Laplacian of an $n$-vertex Eulerian digraph $\Gamma$ gives rise to a finite group $\mathbb{Z}^{n-1}/\mathbb{Z}^{n-1}M$ known as the sandpile (or critical) group $S(\Gamma)$ of $\Gamma$. We determine $S(\Gamma)$ of…

Given a finite simple graph $\Gamma$ on $n$ vertices its complementary prism is the graph $\Gamma\bar{\Gamma}$ that is obtained from $\Gamma$ and its complement $\bar{\Gamma}$ by adding a perfect matching, where each its edge connects two…

A graph is called a GRR if its automorphism group acts regularly on its vertex-set. Such a graph is necessarily a Cayley graph. Godsil has shown that there are only two infinite families of finite groups that do not admit GRRs : abelian…

An Euclidean greedy embedding of a graph is a straight-line embedding in the plane, such that for every pair of vertices $s$ and $t$, the vertex $s$ has a neighbor $v$ with smaller distance to $t$ than $s$. This drawing style is motivated…

A graph is said to be uniquely hamiltonian if it has a unique hamiltonian cycle. For a natural extension of this concept to infinite graphs, we find all uniquely hamiltonian vertex-transitive graphs with finitely many ends, and also discuss…

For a finite connected graph $G$ and a non-empty subset $S$ of its vertices thought of sinks, the so-called critical group (or sandpile group) $C(G, S)$ has been studied for a long time. We present a class of graphs where such an extension…

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. The first deals with connected…

The majority of graphs whose sandpile groups are known are either regular or simple. We give an explicit formula for a family of non-regular multi-graphs called thick cycles. A thick cycle graph is a cycle where multi-edges are permitted.…

We introduce the labelling map and the quasi-free action of a locally compact abelian group on a graph $C^*$-algebra of a row-finite directed graph. Some necessary conditions for embedding the crossed product to an $AF$ algebra are…

The canonical polynomial is an important output of the multivariable topological Poincar\'e series associated with a normal surface singularity. It can be considered as a multivariable polynomial generalization of the Seiberg--Witten…

We define the action of a locally compact group $G$ on a topological graph $E$. This action induces a natural action of $G$ on the $C^*$-correspondence ${\mathcal H}(E)$ and on the graph $C^*$-algebra $C^*(E)$. If the action is free and…