Deficient ‎square ‎graph of ‎finite ‎group

Let $ G $ be a non-abelian finite group and let $ Z(G) $ be the center of $ G $. Associate a graph $ Gamma_{ds}(G) $ (called deficient square graph of $ G $) with $ G $ as follows: ‎t‎ake $ Gsetminus Z(G) $ as the vertices of $ Gamma_{ds}(G) $ and join two distinct vertices $ x $ and $ y $, whenever $ |{ x, y}|^{2} 4 $. ##We want to explore how the graph theoretical properties of $ Gamma_{ds}(G) $ can ‎be‎ effected on the group theoretical properties of $ G $. ##We claim that if $ G $ and $ H $ are two non-abelian finite groups such that $ Gamma_{ds}(G)cong Gamma_{ds}(H) $, then $ |G |= |H |.$