Multiplicative topological indices: Analytical properties and application to random networks

R. Aguilar-Sanchez; J. A. Mendez-Bermudez; Jose M. Rodriguez; Jose M. Sigarreta

paper

Multiplicative topological indices: Analytical properties and application to random networks

arXiv:2306.02511

Abstract

We make use of multiplicative degree-based topological indices $X_{Π} (G)$ to perform a detailed analytical and statistical study of random networks $G = (V (G), E (G))$ . We consider two classes of indices: $X_{Π} (G) = \prod_{u \in V (G)} F_{V} (d_{u})$ and $X_{Π} (G) = \prod_{uv \in E (G)} F_{E} (d_{u}, d_{v})$ , where $uv$ denotes 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. Specifically, we find analytical inequalities involving these multiplicative indices. Also, we apply $X_{Π} (G)$ on three models of random networks: Erdös-Rényi networks, random geometric graphs, and bipartite random networks. We show that $⟨ ln X_{Π} (G) ⟩$ , normalized to the order of the network, scale with the corresponding average degree; here $⟨ \cdot ⟩$ denotes the average over an ensemble of random networks.

19 pages, 7 figures