IN THIS PAPER WE DEFINE THE RELATIVE NON-COMMUTING GRAPH GH, G WHERE G IS A NONABLIAN GROUP AND H A SUBGROUP OF G. WE OBTAIN UPPER BOUNDS FOR DIAMETER AND GIRTH OF THIS GRAPH. WE DISCUSS ABOUT DOMINATING SET AND PLANARITY OF GH, G. MOREOVER, WE EXPLAIN A CONNECTION BETWEEN GH, G AND THE COMMUTATIVITY DEGREE OF G. FURTHERMORE, WE PROVE THAT IF (H1, G1) AND (H2, G2), ARE RELATIVE ISOCLINIC THEN GH1, G1 @GH2, G2 UNDER SPECIAL CONDITION. CONSEQUENT, WE DISCUSS ABOUT THE ENERGY OF GH, G IN SOME SPECIAL CASES. FINALLY WE COMPUTE THE NUMBER OF SPANNING TREES FOR SOME CERTAIN GROUPS.