On the Sombor index of the total graph and the unit graph of commutative rings

TL;DR

This paper studies the Sombor index of total and unit graphs derived from finite rings, providing explicit calculations for specific ring types and general finite local rings.

Contribution

It introduces formulas for the Sombor index of total and unit graphs of certain finite rings, extending previous graph invariants to algebraic structures.

Findings

Explicit Sombor index formulas for $ ext{Z}_n$ with specific $n$

Computed Sombor index for finite local rings

Extended graph invariant analysis to algebraic structures

Abstract

In this paper, we investigate the Sombor index of the total graph and unit graph of $\mathbb{Z}_n$ which is denoted by $T_{\Gamma}(\mathbb{Z}_n)$ and $G(\mathbb{Z}_n)$ respectively for $n \in \{2k, p^{\alpha}, pq, p^2q\}$ where $p$ and $q$ are distinct odd prime numbers such that $p < q$. Moreover, we compute the Sombor index of any finite local ring.