<p>One of the most important pieces of information related to molecular graphs is the determination (when possible) of upper and lower bounds for their corresponding topological indices. Such bounds make it possible to establish the approximate range of the topological indices in terms of molecular structural parameters. The purpose of this paper is to provide new inequalities for the Gutman-Milovanović index, which generalizes important indices such as the general second Zagreb index and the general sum-connectivity index. Moreover, the characterization of extremal graphs with respect to many of these inequalities is obtained. It is a well-known fact that the main application of topological indices is focused on understanding the physicochemical properties of chemical compounds; nevertheless, some additional applications are given to the study of the physicochemical properties of octane isomers. These are compounds that are especially relevant in the chemical and petroleum industries, but are also beginning to be used in the pharmaceutical industry.</p>