Graphs from quadratic forms and vector spaces over finite fields

TL;DR

This paper classifies quadratic forms over finite fields that produce undirected graphs from vector space conditions, and analyzes their structural properties like connectivity and clique size.

Contribution

It identifies specific quadratic forms that generate undirected graphs for all subspaces and studies their structural differences, including connectivity and clique numbers.

Findings

Forms X^2 ± Y^2 produce well-structured, disconnected graphs with large clique numbers.

Family Q_b yields connected graphs with diameter 2 and smaller clique numbers.

Graphs from Q_b are less structured and often have clique numbers o( |V| ).

Abstract

Let $q$ be an odd prime power, let $n\ge 2$, and let $V\subsetneq \mathbb F_{q^n}$ be a proper $\mathbb F_q$-vector subspace. Given a nonzero quadratic form $Q(X,Y)\in \mathbb F_{q^n}[X,Y]$, we consider the graph $\Gamma(Q,V)$ that naturally arises from the condition $Q(X,Y)\in V$. We determine all quadratic forms $Q$ for which $\Gamma(Q,V)$ is undirected for every $V$. Besides the case $Q(x,y)=XY$, studied earlier by the second author, this essentially leads to the forms $X^2\pm Y^2$ and the family $Q_b(X, Y):=X^2+bXY+Y^2, b\ne 0$. We then study connectedness and clique number for the corresponding graphs. Our results reveal a clear contrast between these cases. The graphs $\Gamma(X^2\pm Y^2, V)$ are well structured, disconnected and their clique number can be as large as $\# V$. On the other hand, the family $Q_b$ seems to yield less structured graphs: the graphs are connected (in fact, of diameter $2$) if $\# V\ge q^{3n/4}$ and, in many cases, their clique number is $o(\# V)$. Our proofs are mainly based on character sums, while requiring a few algebraic and combinatorial ideas. We end the paper with some open problems and remarks, including a short discussion of the complementary case where $q$ is even.