The Smith and critical groups of Paley graphs

Abstract. We determine the distribution of the sandpile group (a.k.a. Jacobian) of the Erdős-Rényi random graph G(n, q) as n goes to infinity. Since any particular group appears with asymptotic probability 0 (as we show), it is natural ask for the asymptotic distribution of Sylow p-subgroups of sandpile groups. We prove the distributions of Sylow p-subgroups converge to specific distributions conjectured by Clancy, Leake, and Payne. These distributions are related to, but different from, the Cohen-Lenstra distribution. Our proof involves first finding the expected number of surjections from the sandpile group to any finite abelian group (the "moments" of a random variable valued in finite abelian groups). To achieve this, we show a universality result for the moments of cokernels of random symmetric integral matrices that is strong enough to handle dependence in the diagonal entries. We then show these moments determine a unique distribution despite their p k 2 -size growth.

show abstract

“…The cokernel of the adjacency matrix is called the Smith group, and has been studied e.g. in [CSX14,DJ13].…”

Section: Introductionmentioning

confidence: 99%

The distribution of sandpile groups of random graphs

Wood

2016

J. Amer. Math. Soc.

127

View full text Add to dashboard Cite

show abstract

“…The critical groups of many classes of graphs have been computed. As a couple of nice examples, we mention threshold graphs (work of B. Jacobson [24]) and Paley graphs (D. B. Chandler, P. Sin, and Q. Xiang [9]). Critical groups have been generalized in various ways.…”

Section: Note a Similar Trick Work For The Matrixmentioning

confidence: 99%

Smith normal form in combinatorics

Stanley

2016

Journal of Combinatorial Theory, Series A

View full text Add to dashboard Cite

This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith normal form of random integer matrices. We then give some examples of Smith normal form and diagonal form arising from (1) symmetric functions, (2) a result of Carlitz, Roselle, and Scoville, and (3) the Varchenko matrix of a hyperplane arrangement.

show abstract

mentioning

confidence: 99%

Difference Families, Skew Hadamard Matrices, and Critical Groups of Doubly Regular Tournaments

Pantangi

2019

Electron. J. Combin.

View full text Add to dashboard Cite

In this paper we investigate the structure of the critical groups of doubly-regular tournaments (DRTs) associated with skew Hadamard difference families (SDFs) with one, two, or four blocks. Brown and Ried found that the existence of a skew Hadamard matrix of order $n+1$ is equivalent to the existence of a DRT on $n$ vertices. A well known construction of a skew Hadamard matrix order $n$ is by constructing skew Hadamard difference sets in abelian groups of order $n-1$. The Paley skew Hadamard matrix is an example of one such construction. Szekeres and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with two blocks. Wallis and Whiteman constructed skew Hadamard matrices from skew Hadamard difference families with four blocks. In this paper we consider the critical groups of DRTs associated with skew Hadamard matrices constructed from skew Hadamard difference families with one, two or four blocks. We compute the critical groups of DRTs associated with skew Hadamard difference families with two or four blocks. We also compute the critical group of the Paley tournament and show that this tournament is inequivalent to the other DRTs we considered. Consequently we prove that the associated skew Hadamard matrices are not equivalent.

show abstract

The Smith and critical groups of Paley graphs

Cited by 25 publications

References 14 publications

The distribution of sandpile groups of random graphs

The distribution of sandpile groups of random graphs

Smith normal form in combinatorics

Difference Families, Skew Hadamard Matrices, and Critical Groups of Doubly Regular Tournaments

Contact Info

Product

Resources

About