Shuriken Graphs Arising from Clean Graphs of Rings and Their Properties Relative to Base Graphs

TL;DR

This paper introduces shuriken graphs derived from the structure of rings and their associated idempotent and clean graphs, analyzing their properties and invariants in relation to the base graphs.

Contribution

It defines the shuriken graph operation based on ring structures and studies its properties, including invariants and Hamiltonian and Eulerian characteristics.

Findings

Shuriken graphs' clique and chromatic numbers depend on base graph properties.

The paper characterizes when shuriken graphs are Eulerian or Hamiltonian.

Topological indices of shuriken graphs are analyzed in relation to base graphs.

Abstract

Let $R$ be a finite ring with identity. The idempotent graph $I(R)$ is the graph whose vertex set consists of the non-trivial idempotent elements of $R$, where two distinct vertices $x$ and $y$ are adjacent if and only if $xy = yx = 0$. The clean graph $Cl_2(R)$ is a graph whose vertices are of the form $(e, u)$, where $e$ is a nonzero idempotent element and $u$ is a unit of $R$. Two distinct vertices $(e,u)$ and $(f, v)$ are adjacent if and only if $ef = fe = 0$ or $uv = vu = 1$. The shuriken graph operation is an operation that arises from the structure of the clean graph and depends on the structure of the associated idempotent graph. In this paper, we study the graph obtained from the shuriken operation and examine how its properties depend on those of the base graph. In particular, we investigate several graph invariants, including the clique number, chromatic number, independence number, and domination number. Moreover, we analyze topological indices and characterize Eulerian and Hamiltonian properties of the resulting shuriken graphs in terms of the properties of the base graphs.