Let G be a group. The co-maximal graph of subgroups of G , denoted by ?(G), is a graph whose vertices are nontrivial and proper subgroups of G and two distinct vertices L and K are adjacent in ?(G) if and only if G = LK. In this M. Sc. thesis we study the diameter, clique number, and vertex chromatic number of ?(G). For instance, if G be free abelian group, then diam(?(G)) =? . We investigate the connectivity of a co-maximal graph of a group. We prove that if ?(?(G)) ? ?, then ?(G) is a connected graph and diam(?(G)) ? ?, where ?(?) and diam(?) are the minmum degree and the diameter of a graph ?. We also show that for a nilpotent group G, the graph ?(G) is connected if and only if ?(G) = ? or G?Z_p^? , where ?(G) is the Frattini subgroup of G and Z_n is the cyclic group of order n. We characterize all finite groups whose co-maximal graphs are connected.
