From Sharma-Mittal to von-Neumann Entropy of a Graph

TL;DR

This paper introduces the Sharma-Mittal entropy for graphs, generalizing von-Neumann entropy, and explores its properties, bounds, and applications to various graph types and network models.

Contribution

It defines Sharma-Mittal entropy for graphs, connects it to existing entropies, and analyzes its behavior across different graph structures and network growth models.

Findings

Explicit formulas for cycle, path, and complete graphs.

Derived bounds for Sharma-Mittal and related entropies.

Analyzed entropy in product graphs and network models.

Abstract

In this article, we introduce the Sharma-Mittal entropy of a graph, which is a generalization of the existing idea of the von-Neumann entropy. The well-known R{\'e}nyi, Thallis, and von-Neumann entropies can be expressed as limiting cases of Sharma-Mittal entropy. We have explicitly calculated them for cycle, path, and complete graphs. Also, we have proposed a number of bounds for these entropies. In addition, we have also discussed the entropy of product graphs, such as Cartesian, Kronecker, Lexicographic, Strong, and Corona products. The change in entropy can also be utilized in the analysis of growing network models (Corona graphs), useful in generating complex networks.