We consider two general classes of multiplicative degree-based topological indices (MTIs). 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. 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.This publication is part of the grant PID2019-106433GB-I00 funded by Agencia Estatal de Investigación MCIN/AEI/ 10.13039/501100011033. J.A.M.-B. thanks support from CONACyT (Grant No. 286633), CONACyT-Fronteras (Grant No. 425854), and VIEP-BUAP (Grant No. 100405811-VIEP2023), Mexico. J.M.R. was supported by the Madrid Government (Comunidad de Madrid-Spain) under the Multiannual Agreement with UC3M in the line of Excellence of University Professors (EPUC3M23), and in the context of the V PRICIT (Regional Programme of Research and Technological Innovation)
