In this thesis , we investigate sigma coloring of graphs and its corresponding parameters. For a nontrivial connected graph G , let c : V ( G ) be a vertex coloring of G , where adjacent vertices may be colored the same . For a vertex v of G , let N ( v ) denote the set of vertices adjacent to v . The color sum ( v ) of v is the sum of the colors of the vertices in N ( v ). If ( u ) ? ( v ) for every two adjacent vertices u and v of G , then c is called a sigma coloring of G . The minimum number of colors required in a sigma coloring of a graph G is called its sigma chromatic number ( G ). The sigma chromatic number of a graph G never exceeds its chromatic number ( G ) and for every pair a , b of positive integers with a , there exists a connected graph G with ( G ) = a and ( G ) = b .