Graphs from quadratic forms and vector spaces over finite fields
For mathematicians working on graph theory over finite fields, this provides a classification and structural analysis of a family of graphs, though the results are incremental.
The paper studies graphs derived from quadratic forms over finite fields, determining which forms yield undirected graphs for all subspaces. It finds that graphs from X^2 ± Y^2 are disconnected with large cliques, while those from X^2 + bXY + Y^2 are connected with small cliques under certain conditions.
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 $Γ(Q,V)$ that naturally arises from the condition $Q(X,Y)\in V$. We determine all quadratic forms $Q$ for which $Γ(Q,V)$ 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 $Γ(X^2\pm Y^2, V)$ 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.