论文标题
在二次字符的佩里图上
On the Paley graph of a quadratic character
论文作者
论文摘要
Paley图在二次残基的分布与图理论之间形成了一个很好的联系。这些图具有显着的特性，使它们在数学的多个分支中有用。通常，对于每个质数$ p $，我们可以使用二次和非二次残基Modulo $ p $构建相应的Paley图。因此，Paley图自然与$ p $的Legendre符号相关联，这是导体$ p $的二次dirichlet。在本文中，我们介绍了广义的Paley图。这些图形与一般的二次dirichlet字符相关。然后，我们将提供一些基本属性。特别是，我们明确地描述了它们的频谱。然后，我们使用这些广义的Paley图来构建一些新的Ramanujan图。最后，使用$ L $ functions的特殊值，我们为其Cheeger编号提供了有效的上限。
Paley graphs form a nice link between the distribution of quadratic residues and graph theory. These graphs possess remarkable properties which make them useful in several branches of mathematics. Classically, for each prime number $p$ we can construct the corresponding Paley graph using quadratic and non-quadratic residues modulo $p$. Therefore, Paley graphs are naturally associated with the Legendre symbol at $p$ which is a quadratic Dirichlet character of conductor $p$. In this article, we introduce the generalized Paley graphs. These are graphs that are associated with a general quadratic Dirichlet character. We will then provide some of their basic properties. In particular, we describe their spectrum explicitly. We then use those generalized Paley graphs to construct some new families of Ramanujan graphs. Finally, using special values of $L$-functions, we provide an effective upper bound for their Cheeger number.
