Cyclic Symmetry of Riemann Tensor in Fuzzy Graph Theory

TL;DR

This paper introduces a fuzzy graph-theoretic analog of the Riemann tensor, analyzing its cyclic symmetry and properties, and explores its potential as a model for the Petrov-Penrose classification in differential geometry.

Contribution

It develops a novel fuzzy graph-based model of the Riemann tensor and investigates its properties and similarities to the classical tensor.

Findings

Fuzzy analog satisfies Riemann tensor properties

Expressed as union of fuzzy complete and Levi-Civita graphs

Potential model for Petrov-Penrose classification

Abstract

In this paper, we define a graph-theoretic analog for the Riemann tensor and analyze properties of the cyclic symmetry. We have developed a fuzzy graph-theoretic analog of the Riemann tensor and have analyzed its properties. We have also shown how the fuzzy analog satisfies the properties of the 6X6 matrix of the Riemann tensor by expressing it as a union of the fuzzy complete graph formed by the permuting vertex set and a Levi-Civita graph analog. We have concluded the paper with a brief discussion on the similarities between the properties of the fuzzy graphical analog and the Riemann tensor and how it can be a plausible analogous model for the Petrov-Penrose classification.