To construct a Paley graph, we fix a finite field and consider its elements
as vertices of the Paley graph. Two vertices are connected by an edge if their
difference is a square in the field. We will study some important properties of
the Paley graphs. In particular, we will show that the Paley graphs are
connected, symmetric, and self-complementary. Also we will show that the Paley
graph of order q is (q-1)/2 -regular, and every two adjacent vertices have
(q-5)/4 common neighbors, and every two non-adjacent vertices have q-1/4 common
neighbors, which means that the Paley graphs are strongly regular with
parameters(q,q-1/2,q-5/4, q-1/4). Paley graphs are generalized by many
mathematicians. In the first section of Chapter 3 we will see three examples of
these generalizations and some of their basic properties. In the second section
of Chapter 3 we will define a new generalization of the Paley graphs, in which
pairs of elements of a finite field are connected by an edge if and only if
there difference belongs to the m-th power of the multiplicative group of the
field, for any odd integer m > 1, and we call them the m-Paley graphs. In the
third section we will show that the m-Paley graph of order q is complete if and
only if gcd(m, q - 1) = 1 and when d = gcd(m, q - 1) > 1, the m-Paley graph is
q-1/d -regular. Also we will prove that the m-Paley graphs are symmetric but
not self-complementary. We will show also that the m-Paley graphs of prime
order are connected but the m-Paley graphs of order p^n, n > 1 are not
necessary connected, for example they are disconnected if gcd(m, p^n - 1)
=(p^n-1)/ 2.Comment: Master Thesi