Generalized zero-divisor graph of $*$-rings

Abstract

Let $R$ be a ring with involution $*$ and $Z^*(R)$ denotes the set of all non-zero zero-divisors of $R$. We associate a simple (undirected) graph $Γ'(R)$ with vertex set $Z^*(R)$ and two distinct vertices $x$ and $y$ are adjacent in $Γ'(R)$ if and only if $x^ny^*=0$ or $y^nx^*=0$, for some positive integer $n$. We find the diameter and girth of $Γ'(R)$. The characterizations are obtained for $*$-rings having $Γ'(R)$ a connected graph, a complete graph, and a star graph. Further, we have shown that for a ring $R$, there is an involution on $R\times R$ such that $Γ'(R\times R)$ is disconnected if and only if $R$ is an integral domain.

Figures (8)

• Figure 1: Zero-divisor graphs $\Gamma^*(\mathbb Z_8)$ and $\Gamma^{'}(\mathbb Z_8)$
• Figure 2: $\overline{\Gamma(M_2(\mathbb Z_2))}$
• Figure 3: $\Gamma'(M_2(\mathbb Z_2))$
• Figure 4: $\Gamma'(\mathbb Z_3 \times \mathbb Z_3)$
• Figure 5: $\Gamma^*(M_2(\mathbb Z_2))$

Theorems & Definitions (43)

• Definition 2.1
• Example 2.2
• Remark 2.3
• Theorem 2.4
• proof
• Example 2.5
• Lemma 2.6
• proof
• Lemma 2.7
• proof