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Multiplicative topological indices: Analytical properties and application to random networks

Authors :
Aguilar-Sanchez, R.
Mendez-Bermudez, J. A.
Rodriguez, Jose M.
Sigarreta, Jose M.
Publication Year :
2023

Abstract

We make use of multiplicative degree-based topological indices $X_\Pi(G)$ to perform a detailed analytical and statistical study of random networks $G=(V(G),E(G))$. We consider two classes of indices: $X_\Pi(G) = \prod_{u \in V(G)} F_V(d_u)$ and $X_\Pi(G) = \prod_{uv \in E(G)} F_E(d_u,d_v)$, 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 $X_\Pi(G)$ on three models of random networks: Erd\"os-R\'enyi networks, random geometric graphs, and bipartite random networks. We show that $\left< \ln X_\Pi(G) \right>$, normalized to the order of the network, scale with the corresponding average degree; here $\left< \cdot \right>$ denotes the average over an ensemble of random networks.<br />Comment: 19 pages, 7 figures

Subjects

Subjects :
Mathematics - Combinatorics

Details

Database :
arXiv
Publication Type :
Report
Accession number :
edsarx.2306.02511
Document Type :
Working Paper