We consider two general classes of multiplicative degree-based topological indices (MTIs), denoted by $X_{\mathrm{\Pi },F_V}(G)=\prod _{u\in V(G)}{F}_V(d_u)$ and $X_{\mathrm{\Pi },F_E}(G)=\prod _{uv\in E(G)}{F}_E(d_u,d_v)$, where $uv$ indicates the edge of $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $F_V(x)$ and $F_E(x,y)$ are functions of the vertex degrees. This work has three objectives: First, we follow an analytical approach to deal with a classical topic in the study of topological indices: to find inequalities that relate two MTIs between them, but also to their additive versions $X_{\mathrm{\Sigma }}(G)$. Second, we propose some statistical analysis of MTIs as a generic tool for studying average properties of random networks, extending these techniques for the first time to the context of MTIs. Finally, we perform an innovative scaling analysis of MTIs which allows us to state a scaling law that relates different random graph models.

In 2021, Ivan Gutman introduced the Sombor index, a new vertex-degree-based topological index with significant geometric meaning. This index has shown remarkable growth in research activity in recent years. Following this geometric approach, in this paper we propose several generalizations of the Sombor integral indices. In addition, we study their properties and applications in modeling the enthalpy of vaporization of octane isomers.

The determination of upper and lower bounds for topological indices in molecular graphs provides critical insights into the structural properties of chemical compounds. These bounds facilitate the estimation of the ranges of topological indices based on molecular structural parameters. This study presents novel inequalities for the Gutman-Milovanović index, which generalizes several significant indices such as the first and second Zagreb indices, the Randić index, the harmonic index, the geometric-arithmetic index, the general second Zagreb index, and the general sum-connectivity index. Moreover, we derive and characterize extremal graphs for many of these inequalities. Additionally, we explore the application of the Gutman-Milovanović index in modeling the physicochemical properties of 22 polycyclic aromatic hydrocarbons. Our results demonstrate that the topological index $M_{\alpha ,\beta }$ provides accurate predictions for these properties, with $R^2$ values ranging from 0.9406 to 0.9983, indicating a strong correlation between the index and experimental data. The findings underscore the versatility of $M_{\alpha ,\beta }$ in chemical applications.