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<p>Let <i>n</i> be a nonprime integer. We introduce a new simple undirected graph and denote it by <i>MD</i>(<i>n</i>)<i>.</i> Here, the vertices are the proper divisors of <i>n</i> and two vertices <i>x</i> and <i>y</i> are adjacent if <i>xy</i> divides <i>n.</i> We explore the connectedness of <i>MD</i>(<i>n</i>) and provide detailed calculations for the degree of each vertex. In addition, we focus on a special case <i>n</i> = <i>p</i><sup><i>α</i></sup><i>,</i> where <i>p</i> is a prime positive integer and <i>α</i> ≥ 3 is a positive integer. For these cases, we explicitly determine the chromatic number <i>χ</i> and the clique number <i>ω</i> of <i>MD</i>(<i>n</i>)<i>.</i> Finally, we conclude that <i>χ</i>(<i>MD</i>(<i>n</i>)) = <i>ω</i>(<i>MD</i>(<i>n</i>))<i>.</i></p>

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Integer Divisor Connectivity Graph

  • 期刊论文

  • 发表日期：

    2025年4月1日

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  • M. Jorf,
  • L. Oukhtite

摘要

Let n be a nonprime integer. We introduce a new simple undirected graph and denote it by MD(n). Here, 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 a special case n = pα, where p is a prime positive integer and α ≥ 3 is a positive integer. For these cases, we explicitly determine the chromatic number χ and the clique number ω of MD(n). Finally, we conclude that χ(MD(n)) = ω(MD(n)).