论文标题
PEISERT图和有限字段的超测量功能的完整子图数量
Number of complete subgraphs of Peisert graphs and finite field hypergeometric functions
论文作者
论文摘要
对于prime $ p \ equiv 3 \ pmod {4} $和一个正整数$ t $，让$ q = p = p^{2t} $。令$ g $为有限字段$ \ mathbb {f} _q $的原始元素。 PEISERT图$ p^\ ast（q）$定义为带有顶点的图形$ \ mathbb {f} _q $，其中$ ab $在\ in \ in \ langle g^4 \ rangle \ rangle \ rangle \ cup g \ langle g \ langle g^4 \ rangle $时才仅是边缘。我们提供了一个公式，就有限的字段超几何函数而言，对于$ p^\ ast（q）$中包含的四个订单的完整子图的数量。我们还通过评估某些字符总和来给出$ p^\ ast（q）$中包含的第三个订单的完整子图数的新证明。第四订单的完整子图数量的计算非常乏味，因此我们进一步为PEISERT图中任何订单$ M $的完整子图的数量提供了渐近结果。
For a prime $p\equiv 3\pmod{4}$ and a positive integer $t$, let $q=p^{2t}$. Let $g$ be a primitive element of the finite field $\mathbb{F}_q$. The Peisert graph $P^\ast(q)$ is defined as the graph with vertex set $\mathbb{F}_q$ where $ab$ is an edge if and only if $a-b\in\langle g^4\rangle \cup g\langle g^4\rangle$. We provide a formula, in terms of finite field hypergeometric functions, for the number of complete subgraphs of order four contained in $P^\ast(q)$. We also give a new proof for the number of complete subgraphs of order three contained in $P^\ast(q)$ by evaluating certain character sums. The computations for the number of complete subgraphs of order four are quite tedious, so we further give an asymptotic result for the number of complete subgraphs of any order $m$ in Peisert graphs.
