论文标题
从矢量空间的有限磁场上的浅色图
Paley-like graphs over finite fields from vector spaces
论文作者
论文摘要
由有限场上的众所周知的Paley图及其概括的动机，在本文中，我们探索了由有限场上的向量空间引起的此类图的自然乘法类似物。也就是说，如果$ n \ ge 2 $和$ u \ subsetNeq \ mathbb f_ {q^n} $是$ \ mathbb f_q $ -vector-vector Space，$ g_ {u} $是带有Vertex Set $ v（g_u）= \ m m iance = \ mathb f_ = \ qu_ = \ e Edge = q^n} $（g_u v（g_u）的图形（无方向） b）\ in \ mathbb f_ {q^n}^2 \，| \，a \ ne b，ab \ in U \} $。我们描述了$ g_u $的任意最大集团的结构，并在$ g_u $的集团数字$ω（g_u）$上提供界限。特别是，我们计算了任意$ q $和$ n $的$ω（g_u）$的最大可能值。此外，当$ u \ u \ subsetNeq \ mathbb f_ {q^n} $是任何$ \ mathbb f_q $ -vector-dimension $ d_u \ in \ in \ in \ {1，2，n-1 \} $时，我们将获得$ω（g_u）$的确切值。
Motivated by the well-known Paley graphs over finite fields and their generalizations, in this paper we explore a natural multiplicative-additive analogue of such graphs arising from vector spaces over finite fields. Namely, if $n\ge 2$ and $U\subsetneq \mathbb F_{q^n}$ is an $\mathbb F_q$-vector space, $G_{U}$ is the (undirected) graph with vertex set $V(G_U)=\mathbb F_{q^n}$ and edge set $E(G_U)=\{(a, b)\in \mathbb F_{q^n}^2\,|\, a\ne b, ab\in U\}$. We describe the structure of an arbitrary maximal clique in $G_U$ and provide bounds on the clique number $ω(G_U)$ of $G_U$. In particular, we compute the largest possible value of $ω(G_U)$ for arbitrary $q$ and $n$. Moreover, we obtain the exact value of $ω(G_U)$ when $U\subsetneq \mathbb F_{q^n}$ is any $\mathbb F_q$-vector space of dimension $d_U\in \{1, 2, n-1\}$.
