相关论文: The sandpile groups of graphs of classes CH_n (a_1…

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.…

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 $C_{k_1}, \ldots, C_{k_n}$ 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 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,…

A cyclic subgroup graph of a group $G$ is a graph whose vertices are cyclic subgroups of $G$ and two distinct vertices $H_1$ and $H_2$ are adjacent if $H_1\leq H_2$, and there is no subgroup $K$ such that $H_1<K<H_2$. M.T\u{a}rn\u{a}uceanu…

In this note, we describe a construction that leads to families of graphs whose critical groups are cyclic. For some of these families we are able to give a formula for the number of spanning trees of the graph, which then determines the…

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:…

In this article, we give a partial description of the sandpile group of the cone of the cartesian product of graphs in function of the sandpile group of the cone of their factors. Also, we introduce the concept of uniform homomorphism of…

We show that certain graphs of groups with cyclic edge groups are aTmenable. In particular, this holds when each vertex group is either virtually special or acts properly and semisimply on $\mathbb{H}^n$.

We introduce a group naturally acting on aperiodic necklaces of length $n$ with two colours using the 1--1 correspondences between aperiodic necklaces and irreducible polynomials over the field $\F_2$ of two elements. We notice that this…

We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.

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,…

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…

A sequence $D=(d_1,d_2,\ldots,d_n)$ of non-negative integers is called a graphic sequence if there is a simple graph with vertices $v_1,v_2,\ldots,v_n$ such that the degree of $v_i$ is $d_i$ for $1\leq i\leq n$. Given a graph theoretical…

New criteria for which Cayley graphs of cyclic groups of any order can be completely determined--up to isomorphism--by the eigenvalues of their adjacency matrices is presented. Secondly, a new construction for pairs of nonisomorphic Cayley…

Let $G$ be an $n$-vertex connected graph. A cyclic base ordering of $G$ is a cyclic ordering of all edges such that every cyclically consecutive $n-1$ edges induce a spanning tree of $G$. In this project, we study cyclic base ordering of…

Let G be a finite graph, and let G_n be the n-th iterated cone over G. We study the structure of the critical group of G_n arising in divisor and sandpile theory.

Chain graphs combine directed and undirected graphs and their underlying mathematics combines properties of the two. This paper gives a simplified definition of chain graphs based on a hierarchical combination of Bayesian (directed) and…

This is an expository paper. A $1$-cycle in a graph is a set $C$ of edges such that every vertex is contained in an even number of edges from $C$. E.g., a cycle in the sense of graph theory is a $1$-cycle, but not vice versa. It is easy to…

We determine the number of nonequivalent chord diagrams of order $n$ under the action of two groups, $C_{2n}$, a cyclic group of order $2n$, and $D_{2n}$, a dihedral group of order $4n$. Asymptotic formulas are also established.