Some properties of the complement of the dot product graph of a commutative ring

Articles in Press

Document Type : Research Paper

Authors

1 Retired Faculty, Department of Mathematics, Saurashtra University, Rajkot, Gujarat, India

2 Government Polytechnic, Bhuj, Gujarat, India

10.22034/as.2026.20996.1695

Abstract

For a nonzero commutative ring $A$ with identity and a positive integer $n$, let $R = A\times \cdots\times A$ ($n$ times). The total dot product graph of $R$, denoted by $TD(R)$, is an undirected graph with vertex set equals $R\backslash \{(0, \ldots, 0)\}$ and distinct vertices $x$ and $y$ are adjacent if and only if $x\cdot y = 0$, where $x\cdot y$ is the normal dot product of $x$ and $y$. Let $Z(R)$ denote the set of all zero-divisors of $R$ and let us denote $Z(R)\backslash \{(0, \ldots, 0)\}$ by $Z(R)^{*}$. The zero-divisor dot product graph of $R$, denoted by $ZD(R)$, is the subgraph of $TD(R)$ induced by $Z(R)^{*}$. The graph $TD(R)$ (respectively, $ZD(R)$) was introduced and investigated by Badawi [{\it Comm. Algebra} {\bf 43}(1) (2015), 43-50]. We denote the complement of $TD(R)$ (respectively, $ZD(R)$) by $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$). In this paper, we determine the girth of $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$). Moreover, we characterize $R$ such that $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$) is complemented. Furthermore, if $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$) is connected, then we characterize $R$ such that $(TD(R))^{c}$ (respectively, $(ZD(R))^{c}$) admits a cut edge.
 

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References

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Algebraic Structures and Their Applications

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Available Online from 07 October 2026

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