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Paley graphs form a nice link between the distribution of quadratic residuesand graph theory. These graphs possess remarkable properties which make themuseful in several branches of mathematics. Classically, for each prime number$p$ we can construct the corresponding Paley graph using quadratic andnon-quadratic residues modulo $p$. Therefore, Paley graphs are naturallyassociated with the Legendre symbol at $p$ which is a quadratic Dirichletcharacter of conductor $p$. In this article, we introduce the generalized Paleygraphs. These are graphs that are associated with a general quadratic Dirichletcharacter. We will then provide some of their basic properties. In particular,we describe their spectrum explicitly. We then use those generalized Paleygraphs to construct some new families of Ramanujan graphs. Finally, usingspecial values of $L$-functions, we provide an effective upper bound for theirCheeger number.

On the Paley graph of a quadratic character