Asymptotic enumeration of symmetric integer matrices with uniform row sums

TL;DR

This paper derives the asymptotic count of symmetric zero-diagonal integer matrices with equal row sums, relating to regular multigraphs and contingency tables, as the matrix size grows large.

Contribution

It provides the first asymptotic enumeration formula for symmetric matrices with uniform row sums, extending understanding of regular multigraphs and contingency tables.

Findings

Asymptotic formula derived for large matrices with large row sums

Results applicable to zero diagonal symmetric contingency tables

Conjecture extends validity to all row sums

Abstract

We investigate the number of symmetric matrices of non-negative integers with zero diagonal such that each row sum is the same. Equivalently, these are zero diagonal symmetric contingency tables with uniform margins, or loop-free regular multigraphs. We determine the asymptotic value of this number as the size of the matrix tends to infinity, provided the row sum is large enough. We conjecture that our answer is valid for all row sums.