Critical groups of iterated cones

Goel, Gopal; Perkinson, David

doi:10.1016/j.laa.2019.01.009

Cited by 12 publications

(5 citation statements)

References 4 publications

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“…The following theorem was independently proved by different authors (see, for example, [28], Remark 3,and [32], Theorem 2). Theorem 11.5.…”

Section: Example 113 (The Möbius Laddermentioning

confidence: 93%

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

Mednykh,

Mednykh

2023

Russian Math. Surveys

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The purpose of this survey is to describe invariants of cyclic coverings of graphs. The covered graph is assumed to be fixed, and the cyclic covering group has an arbitrarily large order. A classical example of such a covering is a circulant graph. It covers a one-vertex graph with a prescribed number of loops. More sophisticated objects representing the family of cyclic coverings include $I$-, $Y$-, and $H$-graphs, generalized Petersen graphs, sandwich-graphs, discrete tori, and many others. We present analytic formulae for counting rooted spanning forests and trees in cyclic coverings, establish their asymptotics, and study the arithmetic properties of these quantities. Moreover, in the case of circulant graphs we give exact formulae for computing the Kirchhoff index and present structural theorems for the Jacobians of such graphs. Bibliography: 95 titles.

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“…The following theorem was independently proved by different authors (see, for example, [28], Remark 3,and [32], Theorem 2). Theorem 11.5.…”

Section: Example 113 (The Möbius Laddermentioning

confidence: 93%

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

Mednykh,

Mednykh

2023

Russian Math. Surveys

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show abstract

“…Otherwise n = 5, and we have S = [2,3,4,6,7] with b = 9 and c = 10. We observe 3 is adjacent to 4, 6, and 7, contradicting the fact that it must have degree 2.…”

Section: Spum Of Cyclesmentioning

confidence: 99%

“…Repeatedly joining K 1 to a graph n times, which is equivalent to G + K n , has been called the nth cone over G, and its chip-firing dynamics have been studied by Brown, Morrow, and Zureick-Brown [2] and then by Goel and Perkinson [7].…”

Section: No Additional Edges Are Created Between Two Vertices In L ′mentioning

confidence: 99%

The Spum and Sum-diameter of Graphs: Labelings of Sum Graphs

2021

Preprint

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A sum graph is a finite simple graph whose vertex set is labeled with distinct positive integers such that two vertices are adjacent if and only if the sum of their labels is itself another label. The spum of a graph G is the minimum difference between the largest and smallest labels in a sum graph consisting of G and the minimum number of additional isolated vertices necessary so that a sum graph labeling exists. We investigate the spum of various families of graphs, namely cycles, paths, and matchings. We introduce the sum-diameter, a modification of the definition of spum that omits the requirement that the number of additional isolated vertices in the sum graph is minimal, which we believe is a more natural quantity to study. We then provide asymptotically tight general bounds on both sides for the sum-diameter, and study its behavior under numerous binary graph operations as well as vertex and edge operations. Finally, we generalize the sum-diameter to hypergraphs.

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“…Goel and Perkinson show that this involves understanding when Z/(n + k)Z is a direct summand of cok(nI k + L(G) + 1). This question is analyzed for the path on 4 vertices in [39,Example 5].…”

Section: Subdividing An Edgementioning

confidence: 99%

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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Critical groups of iterated cones

Abstract: Let G be a finite graph, and let Gn be the n-th iterated cone over G. We study the structure of the critical group of Gn arising in divisor and sandpile theory.2010 Mathematics Subject Classification. primary 05C25, secondary 05C76.

Cited by 12 publications

References 4 publications

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

Cyclic coverings of graphs. Counting rooted spanning forests and trees, Kirchhoff index, and Jacobians

The Spum and Sum-diameter of Graphs: Labelings of Sum Graphs

Chip-Firing Games and Critical Groups

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