Integer divisor connectivity graph
UDC 512.5 Let $n$ be a nonprime integer. We introduce a new simple undirected graph and denote it by $MD(n),$ where the vertices are the proper divisors of $n$ and two vertices $x$ and $y$ are adjacent if $xy$ divides $n.$ We explore the connectedness of $MD(n)$ and provide...
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| Datum: | 2025 |
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| Hauptverfasser: | , |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2025
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/7863 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | UDC 512.5
Let $n$ be a nonprime integer. We introduce a new simple undirected graph and denote it by $MD(n),$ where the vertices are the proper divisors of $n$ and two vertices $x$ and $y$ are adjacent if $xy$ divides $n.$ We explore the connectedness of $MD(n)$ and provide detailed calculations for the degree of each vertex. In addition, we focus on the special case where $n = p^{\alpha},$ where $p$ is a prime positive integer and $\alpha\geq 3$ is a positive integer. For these instances, we explicitly determine the chromatic number $\chi$ and the clique number $\omega$ of $MD(n).$ Finally, we conclude that $\chi(MD(n)) = \omega(MD(n)).$ |
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| DOI: | 10.3842/umzh.v76i11.7863 |