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 |.$