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Register a new userThe sandpile group of a trinity and a canonical definition for the planar Bernardi action
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Baker and Wang define the so-called Bernardi action of the sandpile group of a ribbon graph on the set of its spanning trees. This potentially depends on a fixed vertex of the graph but it is independent of the base vertex if and only if the ribbon structure is planar, moreover, in this case the Bernardi action is compatible with planar duality. Earlier, Chan, Church and Grochow and Chan, Glass, Macauley, Perkinson, Werner and Yang proved analogous results about the rotor-routing action. Baker and Wang moreover showed that the Bernardi and rotor-routing actions coincide for plane graphs. We clarify this still confounding picture by giving a canonical definition for the planar Bernardi/rotor-routing action, and also a canonical isomorphism between sandpile groups of planar dual graphs. Our canonical definition implies the compatibility with planar duality via an extremely short argument. We also show hidden symmetries of the problem by proving our results in the slightly more general setting of balanced plane digraphs. Any balanced plane digraph gives rise to a trinity, i.e., a triangulation of the sphere with a three-coloring of the $0$-simplices. Our most important tool is a group associated to trinities, introduced by Cavenagh and Wanless, and a result of a subset of the authors characterizing the Bernardi bijection in terms of a dissection of a root polytope.
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Let $C_t$ be a cycle of length $t$, and let $P_1,ldots,P_t$ be $t$ polygon chains. A polygon flower $F=(C_t; P_1,ldots,P_t)$ is a graph obtained by identifying the $i$th edge of $C_t$ with an edge $e_i$ that belongs to an end-polygon of $P_i$ for $i=1,ldots,t$. In this paper, we first give an explicit formula for the sandpile group $S(F)$ of $F$, which shows that the structure of $S(F)$ only depends on the numbers of spanning trees of $P_i$ and $P_i/ e_i$, $i=1,ldots,t$. By analyzing the arithmetic properties of those numbers, we give a simple formula for the minimum number of generators of $S(F)$, by which a sufficient and necessary condition for $S(F)$ being cyclic is obtained. Finally, we obtain a classification of edges that generate the sandpile group. Although the main results concern only a class of outerplanar graphs, the proof methods used in the paper may be of much more general interest. We make use of the graph structure to find a set of generators and a relation matrix $R$, which has the same form for any $F$ and has much smaller size than that of the (reduced) Laplacian matrix, which is the most popular relation matrix used to study the sandpile group of a graph.
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. Its sandpile group is the direct sum of cyclic groups of orders given by quotients of greatest common divisors of minors of its Laplacian matrix. We show these greatest common divisors can be expressed in terms of monomials in the graphs edge multiplicities.
Let $G$ be a ribbon graph. Matthew Baker and Yao Wang proved that the rotor-routing torsor and the Bernardi torsor for $G$, which are two torsor structures on the set of spanning trees for the Picard group of $G$, coincide when $G$ is planar. We prove the conjecture raised by them that the two torsors disagree when $G$ is non-planar.
Let $C_{k_1}, ldots, C_{k_n}$ be cycles with $k_igeq 2$ vertices ($1le ile n$). 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 order, we obtain a graph, which is called a polygon ring if it can be embedded on the plane; and called a twisted polygon ring if it can be embedded on the M{o}bius band. It is known that the sandpile group of a polygon chain is always cyclic. Furthermore, there exist edge generators. In this paper, we not only show that the sandpile group of any (twisted) polygon ring can be generated by at most three edges, but also give an explicit relation matrix among these edges. So we obtain a uniform method to compute the sandpile group of arbitrary (twisted) polygon rings, as well as the number of spanning trees of (twisted) polygon rings. As an application, we compute the sandpile groups of several infinite families of polygon rings, including some that have been done before by ad hoc methods, such as, generalized wheel graphs, ladders and M{o}bius ladders.
In this article, we consider involutions, called togglings, on the set of independent sets of the Dynkin diagram of type A, or a path graph. We are interested in the action of the subgroup of the symmetric group of the set of independent sets generated by togglings. We show that the subgroup coincides with the symmetric group.
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