Let G be a nite group. The comaximal graph of G, denoted by Г_м (G), is a graph whose vertices are the proper subgroups of G that are not contained in the Frattini subgroup of G and join two distinct vertices H and K, whenever G = hH;Ki. In this paper, we dene an equivalence relation  on V ( Г_м (G)) by taking H  K if and only if their open neighborhoods are the same. We introduce a new graph determined by equivalence classes of V ( Г_м (G)), denoted Г_E (G), as follows. The vertices are V ( Г_E (G)) = f[H]jH 2 V ( Г_м (G))g and two equivalence classes [H] and [K] are adjacent in Г_E (G) if and only if H and K are adjacent in Г_м (G). We will state some basic graph theoretic properties of Г_E (G) and study the relations between some properties of graph Г_м (G) and Г_E (G), such as the chromatic number, clique number, girth and diameter. Moreover, we classify the groups for which Г_E (G) is complete, regular or planar. Among other results, we show that if the number of maximal subgroups of the group G is less or equal than 4, then Г_м (G) and Г_E (G) are perfect graphs.