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Total perfect codes in graphs realized by commutative rings | ||
| Transactions on Combinatorics | ||
| مقاله 1، دوره 11، شماره 4، اسفند 2022، صفحه 295-307 اصل مقاله (455.36 K) | ||
| نوع مقاله: Research Paper | ||
| شناسه دیجیتال (DOI): 10.22108/toc.2021.122946.1727 | ||
| نویسنده | ||
| Rameez Raja* | ||
| Department of Mathematics, National Institute of Technology, Hazratbal-190006, Srinagar, India | ||
| چکیده | ||
| Let $R$ be a commutative ring with unity not equal to zero and let $\Gamma(R)$ be a zero-divisor graph realized by $R$. For a simple, undirected, connected graph $G = (V, E)$, a {\it total perfect code} denoted by $C(G)$ in $G$ is a subset $C(G) \subseteq V(G)$ such that $|N(v) \cap C(G)| = 1$ for all $v \in V(G)$, where $N(v)$ denotes the open neighbourhood of a vertex $v$ in $G$. In this paper, we study total perfect codes in graphs which are realized as zero-divisor graphs. We show a zero-divisor graph realized by a local commutative ring with unity admits a total perfect code if and only if the graph has degree one vertices. We also show that if $\Gamma(R)$ is a regular graph on $|Z^*(R)|$ number of vertices, then $R$ is a reduced ring and $|Z^*(R)| \equiv 0 (mod ~2)$, where $Z^*(R)$ is a set of non-zero zero-divisors of $R$. We provide a characterization for all commutative rings with unity of which the realized zero-divisor graphs admit total perfect codes. Finally, we determine the cardinality of a total perfect code in $\Gamma(R)$ and discuss the significance of the study of total perfect codes in graphs realized by commutative rings with unity. | ||
| کلیدواژهها | ||
| Ring؛ zero-divisor؛ zero-divisor graph؛ perfect code؛ total perfect code | ||
| مراجع | ||
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