Zero-divisor graphs of semirings with no S-vertices

Document Type : Research Paper

Authors

Department of Mathematics and Applied Mathematics‎, ‎University of the Western Cape‎, ‎Private Bag X17‎, ‎Bellville 7535‎, ‎Cape Town‎, ‎South Africa

Abstract

Let $R$ be a commutative semiring (ring) with identity $1 \neq 0$‎. ‎A vertex $a$ in a simple graph $G$ is said to be a Smarandache vertex (or S-vertex for short) provided that there exist three distinct vertices $x$‎, ‎$y$‎, ‎and $b$ (all different from $a$) in $G$ such‎ ‎that $x$---$a$‎, ‎$a$---$b$‎, ‎and $b$---$y$ are edges in $G$‎, ‎but there is no edge between $x$ and $y$‎. ‎In this interdisciplinary subject‎, ‎we investigate‎ ‎the interplay between the algebraic properties of the commutative semirings and their associated zero-divisor graphs‎, ‎denoted by $\Gamma(R)$‎, ‎using the notion of the S-vertices in connection with the nonexistence of S-vertices in $\Gamma(R)$‎. ‎We discuss when $\Gamma(R)$ is a complete bipartite graph‎ ‎together with some of its other graph-theoretic properties and their relation to the nonexistence of S-vertices of $\Gamma(R)$.

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References

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Journal of Algebra and Related Topics
Volume 14, Issue 1
August 2026
Pages 147-157

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