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.
A reduction formula for Waring numbers through generalized Paley graphs
给出了有限域$\mathbb{F}_q$上Waring数$g(k,q)$的一个约化公式.利用广义Paley图$Γ(k,q)$的直径与$g(k,q)$之间的关系，并利用Pearce和Praeger（2019）对笛卡尔可分解的$Γ(k,q)$的刻画，在一定的算术条件下，得到了$p$素数和$a,b,c$正整数的约化公式$$g(\tfrac{p^{ab}-1}{bc},p^{ab}) = b g(\tfrac{p^a-1}{c},p^a)$$.然后，我们找到了应用上述公式的一些算术条件，这些条件允许我们获得许多无限族的显式值的Waring数。最后，我们利用Schmidt和White（2002）给出的约化公式和$2$ -权不可约循环码的特征，找到了$g(k,q)$的显偶数值的无限族.
