Related papers: Chip-firing on graphs of groups

It is known that isomorphisms of graph Jacobians induce cyclic bijections on the associated graphs. We characterize when such cyclic bijections can be strengthened to graph isomorphisms, in terms of an easily computed divisor. The result…

The kernel of the natural projection of a graph product of groups onto their direct product is called the Cartesian subgroup of the graph product. This construction generalises commutator subgroups of right-angled Coxeter and Artin groups.…

We give a formula relating the order of the Brauer group of a surface fibered over a curve over a finite field to the order of the Tate-Shafarevich group of the Jacobian of the generic fiber. The formula implies that the Brauer group of a…

A genus one curve C of degree 5 is defined by the 4 x 4 Pfaffians of a 5 x 5 alternating matrix of linear forms on P^4. We prove a result characterising the covariants for these models in terms of their restrictions to the family of curves…

For any given graph $G$ consider a graph $\widetilde{G}$ which is a cone over graph $G.$ In this paper, we study two important invariants of such a cone. Namely, complexity (the number of spanning trees) and the Jacobian of a graph. We…

Let $\Gamma$ be a finite graph and let $\Gamma_n$ be the "$n$th cone over $\Gamma$" (i.e., the join of $\Gamma$ and the complete graph $K_n$). We study the asymptotic structure of the chip-firing group $\text{Pic}^0(\Gamma_n)$.

We consider a category of continuous Hilbert space representations and a category of smooth Frechet representations, of a real Jacobi group $G$. By Mackey's theory, they are respectively equivalent to certain categories of representations…

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 study the automorphism group of graphons (graph limits). We prove that after an appropriate "standardization" of the graphon, the automorphism group is compact. Furthermore, we characterize the orbits of the automorphism group on…

Let $G$ be a finite group. A number of graphs with the vertex set $G$ have been studied, including the power graph, enhanced power graph, and commuting graph. These graphs form a hierarchy under the inclusion of edge sets, and it is useful…

To any nodal curve $C$ is associated the degree class group, a combinatorial invariant which plays an important role in the compactification of the generalised Jacobian of $C$ and in the construction of the N\'eron model of the Picard…

In the present paper we suggest a simple approach for counting Jacobian group of the $Y$-graph $Y(n; k, l, m).$ In the case $Y(n; 1, 1, 1)$ the structure of the Jacobian group will be find explicitly. Also, we obtain a closed formula for…

Many questions at the core of graph theory can be formulated as questions about certain group-valued flows: examples are the cycle double cover conjecture, Berge-Fulkerson conjecture, and Tutte's 3-flow, 4-flow, and 5-flow conjectures. As…

Let X be a tropical curve (or metric graph), and fix a base point p on X. We define the Jacobian group J(G) of a finite weighted graph G, and show that the Jacobian J(X) is canonically isomorphic to the direct limit of J(G) over all…

The paper describes an explicit combinatorial formula for a harmonic vector for the Laplacian of a directed graph with arbitrary edge weights. This result was motivated by questions from mathematical economics, and the formula plays a…

Given a graph $G=(V, E)$, its generalized Laplacian matrix is given by \[ L(G,X_G)_{u,v}= \begin{cases} x_u&\text{if }u=v,\\ -m_{uv}&\text{if }u\neq v, \end{cases} \] where $X_G=\{x_u\, | \, u\in V(G)\}$ is a set of indeterminates and…

We give algorithms for computing with divisors on projective curves over finite fields, and with their Jacobians, using the algorithmic representation of projective curves developed by Khuri-Makdisi. We show that many desirable operations…

We study harmonic morphisms of graphs as a natural discrete analogue of holomorphic maps between Riemann surfaces. We formulate a graph-theoretic analogue of the classical Riemann-Hurwitz formula, study the functorial maps on Jacobians and…

We study operators on rooted graphs with a certain spherical homogeneity. These graphs are called path commuting and allow for a decomposition of the adjacency matrix and the Laplacian into a direct sum of Jacobi matrices which reflect the…

We derive a formula connecting the orders of the automorphism groups of a finite group and of its covering groups.