  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.
