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Register a new userThe sandpile group of polygon rings and twisted polygon rings
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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.
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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.
We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group $PSL(2,mathbb{Z})$. These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of $PSL(2,mathbb{Z})$.
Curve pseudo-visibility graphs generalize polygon and pseudo-polygon visibility graphs and form a hereditary class of graphs. We prove that every curve pseudo-visibility graph with clique number $omega$ has chromatic number at most $3cdot 4^{omega-1}$. The proof is carried through in the setting of ordered graphs; we identify two conditions satisfied by every curve pseudo-visibility graph (considered as an ordered graph) and prove that they are sufficient for the claimed bound. The proof is algorithmic: both the clique number and a colouring with the claimed number of colours can be computed in polynomial time.
We study a $2 times 2$ matrix equation arising naturally in the theory of Coxeter frieze patterns. It is formulated in terms of the generators of the group $mathrm{PSL}(2,mathbb{Z})$ and is closely related to continued fractions. It appears in a number of different areas, for example, toric varieties. We count its positive solutions, obtaining a series of integer sequences, some known and some new. This extends classical work of Conway and Coxeter proving that the first of these sequences is the Catalan numbers.
During the past three decades fundamental progress has been made on constructing large torsion-free subgroups (i.e. subgroups of finite index) of the unit group $U (Z G)$ of the integral group ring $Z G$ of a finite group $G$. These constructions rely on explicit constructions of units in $Z G$ and proofs of main results make use of the description of the Wedderburn components of the rational group algebra $Q G$. The latter relies on explicit constructions of primitive central idempotents and the rational representations of $G$. It turns out that the existence of reduced two degree representations play a crucial role. Although the unit group is far from being understood, some structure results on this group have been obtained. In this paper we give a survey of some of the fundamental results and the essential needed techniques.
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