Multiplicative topological indices: Analytical properties and application to random networks

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.