Abstarct: Let R be a ring. The unit graph of R, denoted by G(R), is the simple graph defined on all elements of R, and two distinct vertices x and y are adjacent if and only if x+y is aunit of R. The diameter of a simple graph G, denoted by diam(G), is the longest distance between all pairs of vertics of the graph G. In this thesis, we prove that for each integer n≥1, there exist a rings R such that n≤diam(G(R))≤2n. we also show that diam(G(R))∈{1,2,3,∞} for a ring R with R/J(R) self-injective and classify all those rings with diam(G(R))=1,2,3 and ∞ respectively. Furthermore, if R is a ring, then the following are equivalent: • diam(G(R ̅))<diam(G(R)). • R is a local ring with J(R)≠0 and 2∈J(R) • diam(G(R))=2 and diam(G(R ̅))=1. Also for a right self-injective ring R, the following conditions are equivalent: • Every element of R is a sum of two units. • Identity of R is a sum of two units.. • R has no fact or ring isomorphic to Z_2. For any nonzero regular right self-injective ring R, where usn(M_2 (R))=2. In particular, usn(R)=2 for any purely infinitive regular right self-injective ring R. Finally we show a nonzero regular right self-injective ring R has a ring decomposition R=S×T where usn(S)=1 or 2 and T is an abelian regular right self-injective ring.