Let R be a ring with involution <InlineEquation ID="IEq1"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>. The <InlineEquation ID="IEq2"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-symmetric graph of R is a simple graph with vertex set as the set of all nonzero zero-divisors of R and two distinct vertices x and y are adjacent if <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$xy=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>x</mi> <mi>y</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> or <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$yx=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <mi>x</mi> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$yx^=0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>y</mi> <msup> <mi>x</mi> <mo>∗</mo> </msup> <mo>=</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. The <InlineEquation ID="IEq6"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-symmetric graph is a generalization of the well known zero-divisor graph of R. In this paper, we investigate the interplay between the ring-theoretic properties of <InlineEquation ID="IEq7"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-ring R and the graph-theoretic properties of <InlineEquation ID="IEq8"> <EquationSource Format="TEX"></EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-symmetric graph of R.

Graph invariants of the symmetric graph of a ring with involution

期刊论文
发表日期：

2026年7月23日

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W. M. Fakieh,
T. Asir,
A. M. Anto

摘要

Let R be a ring with involution $*$ $*$ . The $*$ $*$ -symmetric graph of R is a simple graph with vertex set as the set of all nonzero zero-divisors of R and two distinct vertices x and y are adjacent if $xy=0$ $x y = 0$ or $yx=0$ $y x = 0$ and $yx^*=0$ $y x^{*} = 0$ . The $*$ $*$ -symmetric graph is a generalization of the well known zero-divisor graph of R. In this paper, we investigate the interplay between the ring-theoretic properties of $*$ $*$ -ring R and the graph-theoretic properties of $*$ $*$ -symmetric graph of R.