Let G be a graph which is neither empty nor complete. Then G is said to be a strongly regular graph (SRG) with parameters (v, k, ? , µ ) if G is a k-regular graph on n vertices in which every pair of adjacent vertices has ? common neighbors and every pair of non-adjacent vertices has µ common neighbors. One of the most important family of regular graphs is the family of strongly regular graphs which has so many beautiful properties. One of these well-known properties is the fact that the eigenvalues of all SRGs depend only the parameters of these graphs. There are many SRGs arising from combinatorial concepts such as orthogonal arrays, Latin squares, conference matrices, designs and geometric graphs. In this thesis, the following generalizations of strongly regular graphs has been studied. A graph G is a Deza graph if it is regular and the number of common neighbors of two distinct vertices takes on one of two values (not necessarily depending on the adjacency of the two vertices). Haemers, Erickson, Fernando, Hard and Hemmeter introduced several ways to construct Deza graphs, and develop some basic theory. They also gave list all diameter two Deza graphs which are not strongly regular and have at most 13 vertices. The Friendship Theorem states that if any two people in a party have exactly one common friend, then there exists a politician who is a friend of everybody. Let ? be any nonnegative integer and µ be any positive integer. Suppose each pair of friends have exactly ? common friends and each pair of strangers have exactly µ common friends in a party. The corresponding graph is a generalization of strongly regular graphs obtained by relaxing the regularity property on vertex degrees. Gera and Shen proved that either everyone has exactly the same number of friends or there exists a politician who is a friend of everybody. A k-regular graph on v vertices is a divisible design graph (DDG for short) with parameters(v,k, ? 1 , ? 2 , m, n) whenever the vertex set can be partitioned into m