Chip-firing groups of iterated cones

Brown, Morgan V.; Morrow, Jackson S.; Zureick-Brown, David

doi:10.1016/j.laa.2018.06.023

Cited by 8 publications

(9 citation statements)

References 16 publications

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“…Remark 2. To see that Corollary B of [2] is equivalent to the result stated in Part 3, note that in [2], the orders of H n and K(G n ) are stated in terms of the characteristic polynomial of the endomorphism of ker δ obtained from our Laplacian by restriction. Calling that characteristic polynomial P G , we have P G (x) = p L (x)/x.…”

Section: Introductionmentioning

confidence: 94%

“…The question of the structure of K(G n ) was addressed previously in [2]. In that paper, Theorem A provides a short exact sequence for K(G n ) (cf.…”

Section: Introductionmentioning

confidence: 99%

“…where H n is a group of order |p L (−n)|/n. Question 1.2 of [2] asks when the exact sequence in Corollary 4 splits. By Theorem 1, (Z/(n + k)Z) n−2 always splits off of K(G n ), and the exact sequence of Theorem A splits exactly when Z/(n + k)Z is a direct summand of cok(nI k + L + 1).…”

Section: Introductionmentioning

confidence: 99%

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Critical groups of iterated cones

Goel

Perkinson

2019

Linear Algebra and its Applications

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Section: Introductionmentioning

confidence: 94%

“…The question of the structure of K(G n ) was addressed previously in [2]. In that paper, Theorem A provides a short exact sequence for K(G n ) (cf.…”

Section: Introductionmentioning

confidence: 99%

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Critical groups of iterated cones

Goel

Perkinson

2019

Linear Algebra and its Applications

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“…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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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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Chip-firing groups of iterated cones

Abstract: Let Γ be a finite graph and let Γ n be the "nth cone over Γ" (i.e., the join of Γ and the complete graph K n ). We study the asymptotic structure of the chip-firing group Pic 0 (Γ n ).

Cited by 8 publications

References 16 publications

Critical groups of iterated cones

Critical groups of iterated cones

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

Chip-Firing Games and Critical Groups

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