In this paper, we solved some open problems on general Gutman index. More precisely, we characterized unicyclic graphs with extremal general Gutman index for some $a$ and $b$. We presented a sharp bound on general Gutman index of $G$ in terms of order and vertex connectivity of $G$. Also, we obtained some bounds on general Gutman index in terms of order, general Randić index, diameter, and independence number of graph $G$. In addition, QSPR analysis on various anticancer drug structures was carried out to relate their physicochemical properties with the general Gutman index of the structure for some $a$ and $b$.