Regular bipartite multigraphs have many (but not too many) symmetries

TL;DR

This paper investigates the symmetry properties of regular bipartite multigraphs, determining bounds on their automorphism groups, and explores their connections to contingency tables and set partitions, revealing most have limited symmetries.

Contribution

It explicitly computes the minimum and maximum automorphism group sizes for certain bipartite graphs and analyzes the typical symmetry structure when one parameter is large.

Findings

Almost all such graphs have automorphism groups fixing vertices pointwise.

The automorphism groups are generally much smaller than the maximum possible.

Most graphs exhibit limited symmetry, with automorphism groups of small order.

Abstract

Let $k$ and $l$ be integers, both at least 2. A $(k,l)$-bipartite graph is an $l$-regular bipartite multigraph with coloured bipartite sets of size $k$. Define $\chi(k,l)$ and $\mu(k,l)$ to be the minimum and maximum order of automorphism groups of $(k,l)$-bipartite graphs, respectively. We determine $\chi(k,l)$ and $\mu(k,l)$ for $k\geq 8$, and analyse the generic situation when $k$ is fixed and $l$ is large. In particular, we show that almost all such graphs have automorphism groups which fix the vertices pointwise and have order far less than $\mu(k,l)$. These graphs are intimately connected with both contingency tables with uniform margins and uniform set partitions; we examine the uniform distribution on the set of $k\times k$ contingency tables with uniform margin $l$, showing that with high probability all entries stray far from the mean. We also show that the symmetric group acting on uniform set partitions is non-synchronizing.