Related papers: The sandpile group of a polygon flower
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…
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.…
The sandpile group of a connected graph is the group of recurrent configurations in the abelian sandpile model on this graph. We study the structure of this group for the case of regular trees. A description of this group is the following:…
Sandpile groups are a subtle graph isomorphism invariant, in the form of a finite abelian group, whose cardinality is the number of spanning trees in the graph. We study their group structure for graphs obtained by attaching a cone vertex…
Let $G$ be a finite, connected, simple graph. The critical group $K(G)$, also known as the sandpile group, is the torsion subgroup of the cokernel of the graph Laplacian $\operatorname{cok}(L)$. We investigate a family of graphs with…
The sandpile group of a graph is a well-studied object that combines ideas from algebraic graph theory, group theory, dynamical systems, and statistical physics. A graph's sandpile group is part of a larger algebraic structure on the graph,…
A wired tree is a graph obtained from a tree by collapsing the leaves to a single vertex. We describe a pair of short exact sequences relating the sandpile group of a wired tree to the sandpile groups of its principal subtrees. In the case…
The cyclic graph $\Gamma(S)$ of a semigroup $S$ is the simple graph whose vertex set is $S$ and two vertices $x, y$ are adjacent if the subsemigroup generated by $x$ and $y$ is monogenic. In this paper, we classify the semigroup $S$ such…
The article considers the procedure of connection of graphs to the edges of a cyclic graph and its influence on the sandpile group of the graph thus obtained. A series of classes of graphs CH_n(a_1,...,a_n) is defined. Recurrent and…
The sandpile group of a connected graph $G$, defined to be the torsion part of the cokernel of the graph Laplacian, is a subtle graph invariant with combinatorial, algebraic, and geometric descriptions. Extending and improving previous…
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 sandpile group of a connected graph is a finite abelian group whose cardinality is the number of spanning trees in the graph. We compute the spanning tree number and sandpile group structure for the cone over a bi-coconut tree,…
We compute the sandpile groups of families of planar graphs having a common weak dual by evaluating the indeterminates of the critical ideals of the weak dual at the lengths of the cycles bounding the interior faces. This method allow us to…
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…
We consider cubic graphs formed with $k \geq 2$ disjoint claws $C_i \sim K_{1, 3}$ ($0 \leq i \leq k-1$) such that for every integer $i$ modulo $k$ the three vertices of degree 1 of $\ C_i$ are joined to the three vertices of degree 1 of…
The abelian sandpile models feature a finite abelian group $G$ generated by the operators corresponding to particle addition at various sites. We study the canonical decomposition of $G$ as a product of cyclic groups $G = Z_{d_1} \times…
The Ehrhart ring of the edge polytope $\mathcal{P}_G$ for a connected simple graph $G$ is known to coincide with the edge ring of the same graph if $G$ satisfies the odd cycle condition. This paper gives for a graph which does not satisfy…
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…
A celebrated result of Seymour is that all regular matroids are built up from graphic matroids, cographic matroids, and a specific 10 element rank 5 matroid called $R_{10}$. In this article, we give a simple description of chip-firing on…
In this paper, we study the independence polynomial $P_G(x)$ of a finite simple graph $G$, with emphasis on the evaluation at $x=-1$, symmetry, and its connection with the $h$-polynomial of the edge ideal of $G$. For big star graphs, we…