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L(2,1)-labeling of some zero-divisor graphs associated with commutative rings | ||
| Communications in Combinatorics and Optimization | ||
| مقالات آماده انتشار، پذیرفته شده، انتشار آنلاین از تاریخ 04 آذر 1402 اصل مقاله (1.51 M) | ||
| نوع مقاله: Original paper | ||
| شناسه دیجیتال (DOI): 10.22049/cco.2023.28810.1730 | ||
| نویسندگان | ||
| Annayat Ali؛ Rameez Raja* | ||
| Department of Mathematics, National Institute of Technology Srinagar | ||
| چکیده | ||
| Let $\mathcal G = (\mathcal V, \mathcal E)$ be a simple graph, an $L(2,1)$-labeling of $\mathcal G$ is an assignment of labels from non-negative integers to vertices of $\mathcal G$ such that adjacent vertices get labels which differ by at least by two, and vertices which are at distance two from each other get different labels. The $\lambda$-number of $\mathcal G$, denoted by $\lambda(\mathcal G)$, is the smallest positive integer $\ell$ such that $\mathcal G$ has an $L(2,1)$-labeling with all labels as members of the set $\{ 0, 1, \dots, \ell \}$. The zero-divisor graph of a finite commutative ring $R$ with unity, denoted by $\Gamma(R)$, is the simple graph whose vertices are all zero divisors of $R$ in which two vertices $u$ and $v$ are adjacent if and only if $uv = 0$ in $R$. In this paper, we investigate $L(2,1)$-labeling of some zero-divisor graphs. We study the \textit{partite truncation}, a graph operation that allows us to obtain a reduced graph of relatively small order from a graph of significantly larger order. We establish the relation between $\lambda$-numbers of the graph and its partite truncated one. We make use of the operation \textit{partite truncation} to contract the zero-divisor graph of a reduced ring to the zero-divisor graph of a Boolean ring. | ||
| کلیدواژه‌ها | ||
| Zero-divisor graph؛ L(2,1)-labeling؛ λ -number؛ partite truncation | ||
| مراجع | ||
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