A graph is a set of vertices with a given adjacencies between them . A dominating set in a graph is a set of vertices , such that every vertex not belonging to this set is adjacent to some vertices in it . In this thesis we study a function on graphs , denoted by “ Gamma ” , representing vectorially the domination number of a graph , in a way similar to that in which the Lov`asz Theta function represents the independence number of a graph . some lower and upper bounds on Gamma , formulated in terms of known domination and algebraic parameters of the graph . These bounds are used to calculate the value of Gamma for cycles and for trees . For this purpose a new variant of the Gamma function is introduced and it is conjectured that to be equal to the original function . Among other properties of the Gamma function that are proved it is shown that , this variant of the Gamma function satisfies a well known conjecture of Vizing on the domination number of graph products . moreover , some are proved results concerning the behavior of when some of the vertices of are duplicated . The results here depend on a new result regarding the behavior of the largest eigenvalue of the laplacian matrix of the graph when some of the vertices of the graph are duplicated . Finally , some applications of Gamma are given , among which are generalizations of results by Furedi , Kahn and Seymour concerning the edge chromatic number of bipartite uniform hypergraphs .