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Zero-divisor graphs of semirings with no S-vertices | ||
| Journal of Algebra and Related Topics | ||
| مقالات آماده انتشار، اصلاح شده برای چاپ، انتشار آنلاین از تاریخ 22 آذر 1404 اصل مقاله (167.48 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22124/jart.2025.29182.1740 | ||
| نویسندگان | ||
| E. Mehdi-Nezhad* ؛ K. O. E. Hassan | ||
| Department of Mathematics and Applied Mathematics, University of the Western Cape, Private Bag X17, Bellville 7535, Cape Town, South Africa | ||
| چکیده | ||
| 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 $\Gamma(R)$. | ||
| کلیدواژهها | ||
| Complete bipartite graph؛ Weakly perfect graph؛ $r$-partite graph؛ Smarandache vertex (S-vertex) of a graph؛ Smarandache zero-divisor | ||
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