A New Family of Algebraically Defined Graphs With Small Automorphism Group

TL;DR

This paper introduces a new family of algebraically defined bipartite graphs over finite fields with small automorphism groups, demonstrating that certain graph isomorphisms cannot be simplified through function replacements.

Contribution

It constructs specific graphs with minimal automorphism groups and proves that they cannot be simplified via function substitution, answering a key question in algebraic graph theory.

Findings

The automorphism group of the constructed graphs has order p.

These graphs serve as counterexamples to a general simplification hypothesis.

The graphs are defined over finite fields with prime characteristic p, where p ≡ 1 mod 3.

Abstract

Let $p$ be an odd prime, $q=p^e$, $e\ge 1$, and $\mathbb{F} = \mathbb{F_q}$ denote the finite field of $q$ elements. Let $f: \mathbb{F}^2\to \mathbb{F}$ and $g: \mathbb{F}^3\to \mathbb{F}$ be functions, and let $P$ and $L$ be two copies of the 3-dimensional vector space $\mathbb{F}^3$. Consider a bipartite graph $\Gamma _\mathbb{F} (f, g)$ with vertex partitions $P$ and $L$ and with edges defined as follows: for every $(p)=(p_1,p_2,p_3)\in P$ and every $[l]= [l_1,l_2,l_3]\in L$, $\{(p), [l]\} = (p)[l]$ is an edge in $\Gamma _\mathbb{F} (f, g)$ if $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = g(p_1,p_2,l_1).$$ Given $\Gamma _\mathbb{F} (f, g)$, is it always possible to find a function $h:\mathbb{F}^2\to \mathbb{F}$ such that the graph $\Gamma _\mathbb{F} (f, h)$ with the same vertex set as $\Gamma _\mathbb{F} (f, g)$ and with edges $(p)[l]$ defined in a similar way by the system $$p_2+l_2 =f(p_1,l_1) \;\;\;\text{and}\;\;\; p_3 + l_3 = h(p_1,l_1),$$ is isomorphic to $\Gamma _\mathbb{F} (f, g)$ for infinitely many $q$? In this paper we show that the answer to the question is negative and the graphs $\Gamma_{\mathbb{F}_p}(p_1\ell_1, p_1\ell_1p_2(p_1 + p_2 + p_1p_2))$ provide such an example for $p \equiv 1 \pmod{3}$. Our argument is based on proving that the automorphism group of these graphs has order $p$, which is the smallest possible order of the automorphism group of graphs of the form $\Gamma_{\mathbb{F}}(f, g)$.