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

TL;DR

The paper studies the $(t,n)$-shuriken graph operation $Shu^t_n(G)$, arising from clean graphs and idempotent graphs associated with finite rings, and analyzes how classical graph invariants and topological indices behave under this construction. It derives explicit formulas and bounds for invariants such as the clique number $ω$, chromatic number $χ$, independence number $α$, domination number $γ$, and Zagreb indices $M_1$, $M_2$, expressed in terms of the base graph $G$’s parameters, and establishes Hamiltonian and Eulerian criteria for the shuriken graphs. Notably, it proves $ω(Shu^t_n(G)) = |V(G)|+1$ when $n=t$, and $ω(Shu^t_n(G)) = ext{max}igligl vert |V(G)|+1, 2ω(G) igr vertigr brace$ when $n-t>0$, provides exact or tight bounds for $χ$, and gives $α(Shu^t_n(G))$ and $γ(Shu^t_n(G))$ in terms of $G$’s parameters; it also gives closed forms for $M_1(Shu^t_n(G))$ and $M_2(Shu^t_n(G))$. The work shows that if $G$ is semi-Hamiltonian or Hamiltonian, then $Shu^t_n(G)$ is Hamiltonian for all admissible $(t,n)$, and characterizes Eulerian cases as $t=n$, $|V(G)|$ even with $G$ Eulerian or $n$ odd. Overall, the results demonstrate that the shuriken construction encodes base-graph structure into rich graph properties, linking algebraic graph constructions with classical invariants and enabling further exploration of algebraically motivated graph transformations.

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.

Figures (1)

• Figure 1: Graphs $Shu^2_{4}(P_3)$ and $Shu^8_{16}(K_2)$

Theorems & Definitions (21)

• Definition 1: djuang
• Theorem 1
• proof
• Proposition 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof
• Theorem 4