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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| Date: | 2025 |
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| Main Authors: | , |
| Format: | Article |
| Language: | English |
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Institute of Mathematics, NAS of Ukraine
2025
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/7863 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860512788184563712 |
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| author | Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. |
| author_facet | Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. |
| author_sort | Jorf, M. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2025-08-09T14:51:14Z |
| description | 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 $\chi(MD(n)) = \omega(MD(n)).$ |
| doi_str_mv | 10.3842/umzh.v76i11.7863 |
| first_indexed | 2026-03-24T03:34:21Z |
| format | Article |
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| id | umjimathkievua-article-7863 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T03:34:21Z |
| publishDate | 2025 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
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| spelling | umjimathkievua-article-78632025-08-09T14:51:14Z Integer divisor connectivity graph Integer divisor connectivity graph Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. divisor, graph, prime number. 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 $\chi(MD(n)) = \omega(MD(n)).$ УДК 512.5 Граф зв'язності цілочислового дільника Нехай $n$ – непросте ціле число. Введено новий простий неорієнтований граф, який позначено $MD(n).$ Вершини цього графа є власними дільниками $n.$ Крім того, дві вершини $x$ і $y$ є суміжними, якщо $xy$ ділить $n.$ Досліджено зв’язність $MD(n)$ і наведено детальні розрахунки для степеня кожної вершини. Крім того, зосереджено увагу на окремому випадку, коли $n = p^{\alpha},$ де $p$ – просте натуральне число, а $\alpha\geq 3$ – натуральне число. Для таких випадків явно визначено хроматичне число $\chi$ і клікове число $\omega$ для $MD(n),$ а також встановлено, що $\chi(MD(n)) = \omega(MD(n)).$  Institute of Mathematics, NAS of Ukraine 2025-08-06 Article Article https://umj.imath.kiev.ua/index.php/umj/article/view/7863 10.3842/umzh.v76i11.7863 Ukrains’kyi Matematychnyi Zhurnal; Vol. 76 No. 11 (2024); 1621 - 1628 Український математичний журнал; Том 76 № 11 (2024); 1621 - 1628 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/7863/10281 Copyright (c) 2024 mohamed jorf |
| spellingShingle | Jorf, M. Oukhtite, L. Jorf, M. Oukhtite, L. Integer divisor connectivity graph |
| title | Integer divisor connectivity graph |
| title_alt | Integer divisor connectivity graph |
| title_full | Integer divisor connectivity graph |
| title_fullStr | Integer divisor connectivity graph |
| title_full_unstemmed | Integer divisor connectivity graph |
| title_short | Integer divisor connectivity graph |
| title_sort | integer divisor connectivity graph |
| topic_facet | divisor graph prime number. |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/7863 |
| work_keys_str_mv | AT jorfm integerdivisorconnectivitygraph AT oukhtitel integerdivisorconnectivitygraph AT jorfm integerdivisorconnectivitygraph AT oukhtitel integerdivisorconnectivitygraph |