Sombor index of clean graphs

TL;DR

This paper investigates the Sombor index of the clean graph of a ring, specifically focusing on the subgraph related to the integers modulo n, providing new insights into its mathematical properties.

Contribution

The paper introduces the calculation of the Sombor index for the clean graph of rings, particularly for the subgraph of integers modulo n, which is a novel exploration in graph theory and algebra.

Findings

Derived formulas for the Sombor index of Cl_2(Z_n) for various n

Identified patterns and properties of the Sombor index in these graphs

Extended understanding of graph invariants in algebraic structures

Abstract

Let $G = (V, E)$ be a graph with the vertex set $V (G)$ and edge set $E(G)$. The Sombor index of $G$, $SO(G)$, is defined as $\sum_{uv\in E(G)} \sqrt{deg(u)^2 + deg(v)^2}$, where $deg(u)$ is the degree of vertex $u$ in $V (G)$. The clean graph of a ring R, denoted by $Cl(R)$, is a graph with vertex set $\{(e, u) : e \in Id(R), u \in U(R)\}$ and two distinct vertices $(e, u)$ and$(f, v)$ are adjacent if and only if $ef = 0$ or $uv = 1$ ($Id(R)$ and $U(R)$ are the sets of idempotents and unit elements of R, respectively). The induced subgraph on $\{(e, u) : e \in Id^{*}(R), u \in U(R)\}$ is denoted by $Cl_2(R)$. In this paper, $SO(Cl2(\mathbb{Z}_n))$, for different values of the positive integer $n$, is investigated.