Two-Vertex Generators of Jacobians of Graphs

Let C t be a cycle of length t, and let P 1 , . . . , P t be t polygon chains. A polygon flower F = (C t ; P 1 , . . . , P t ) is a graph obtained by identifying the ith edge of C t with an edge e i that belongs to an end-polygon of P i for i = 1, . . . , 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, . . . , 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.

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“…[13] Let G be a graph. Then e ∈ E(G) is a generator of the sandpile group S(G) if and only if gcd(τ (G), τ (G/e)) = 1.…”

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confidence: 99%

The sandpile group of a polygon flower

Chen

Mohar

2019

Discrete Applied Mathematics

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“…They propose a general criterion for when a graph G has such a generator. This conjecture was proven in [15].…”

Section: Generators Of Critical Groupsmentioning

confidence: 81%

Chip-Firing Games and Critical Groups

Glass

Kaplan

2020

Foundations for Undergraduate Research in Mathematics

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In this note we introduce a finite abelian group that can be associated to any finite connected graph. This group can be defined in an elementary combinatorial way in terms of chip-firing operations, and has been an object of interest in combinatorics, algebraic geometry, statistical physics, and several other areas of mathematics. We will begin with basic definitions and examples and develop a number of properties that can be derived by looking at this group from different angles. Throughout, we will give exercises, some of which are straightforward and some of which are open questions. We will also attempt to highlight some of the many contributions to this area made by undergraduate students.Suggested prerequisites. The basic definitions and themes of this note should be accessible to any student with some knowledge of linear algebra and group theory. As we go along, deeper understanding of graph theory, abstract algebra, and algebraic geometry will be of use in some sections.

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The sandpile group of a polygon flower

Chen

Mohar

2019

Preprint

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Two-Vertex Generators of Jacobians of Graphs

Cited by 3 publications

References 9 publications

The sandpile group of a polygon flower

The sandpile group of a polygon flower

Chip-Firing Games and Critical Groups

The sandpile group of a polygon flower

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