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.

翻译：设 $q$ 为奇素数幂，$n\ge 2$，且 $V\subsetneq \mathbb F_{q^n}$ 是 $\mathbb F_q$ 的一个真向量子空间。给定一个非零二次型 $Q(X,Y)\in \mathbb F_{q^n}[X,Y]$，我们考虑由条件 $Q(X,Y)\in V$ 自然导出的图 $Γ(Q,V)$。我们确定了所有使得 $Γ(Q,V)$ 对每个 $V$ 均为无向图的二次型 $Q$。除了此前由第二作者研究的 $Q(x,y)=XY$ 情形外，这本质上归结为形式 $X^2\pm Y^2$ 以及族 $Q_b(X, Y):=X^2+bXY+Y^2, b\ne 0$。随后，我们研究了相应图的连通性和团数。我们的结果揭示了这些情形之间的显著对比：图 $Γ(X^2\pm Y^2, V)$ 结构明确、不连通，其团数可达 $\# V$；而族 $Q_b$ 似乎产生结构较弱的图：当 $\# V\ge q^{3n/4}$ 时，这些图是连通的（事实上直径为 $2$），且在多数情况下其团数为 $o(\# V)$。我们的证明主要基于特征和，同时辅以少量代数和组合思想。论文最后给出了一些未解决问题与评注，包括对 $q$ 为偶数情形的简短讨论。