Counting primitive subsets and other statistics of the divisor graph of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e253" altimg="si6.svg"><mml:mrow><mml:mo>{</mml:mo><mml:mn>1</mml:mn><mml:mo>,</mml:mo><mml:mn>2</mml:mn><mml:mo>,</mml:mo><mml:mo>…</mml:mo><mml:mo>,</mml:mo><mml:mi>n</mml:mi><mml:mo>}</mml:mo></mml:mrow></mml:math>

McNew, Nathan

doi:10.1016/j.ejc.2020.103237

Cited by 8 publications

(13 citation statements)

References 10 publications

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“…Note that log(2/ǫ i ) = log 2+i log 10, so that an expression of magnitude 1/ log(2/ǫ i ) is much larger than ǫ i−1 when i ≥ 4, so we have (8).…”

Section: 3mentioning

confidence: 99%

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Coprime permutations

Pomerance¹

2022

Preprint

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“…Note that log(2/ǫ i ) = log 2+i log 10, so that an expression of magnitude 1/ log(2/ǫ i ) is much larger than ǫ i−1 when i ≥ 4, so we have (8).…”

Section: 3mentioning

confidence: 99%

“…Problems such as the longest possible cycle in such permutations, the minimum number of disjoint cycles, etc. were studied in [10], [4], [8] and elsewhere. The enumeration problems have not been well-studied, though the first one has an OEIS page: A320843.…”

Section: Anti-coprime Permutationsmentioning

confidence: 99%

Coprime permutations

Pomerance¹

2022

Preprint

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“…[3][4][5][6]). Moreover, there are many other definitions that connect the two theories of graph and ring (e.g., [7][8][9]). In 2021, Mohammad and Shuker [10] defined the idempotent divisor graph of commutative ring R with identity 1 = 0.…”

Section: Introductionmentioning

confidence: 99%

Vertex and region colorings of planar idempotent divisor graphs of commutative rings.

Authman

2022

ijcsm

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The idempotent divisor graph of a commutative ring R is a graph with vertices set in R* = R-{0}, and any distinct vertices x and y are adjacent if and only if x.y = e, for some non-unit idempotent element e2 = e ϵ R, and is denoted by Л(R). The purpose of this work is using some properties of ring theory and graph theory to find the clique number, the chromatic number and the region chromatic number for every planar idempotent divisor graphs of commutative rings. Also we show the clique number is equal to the chromatic number for any planar idempotent divisor graph. Among other results we prove that: Let Fq, Fpa are fieldes of orders q and pa respectively, where q=2 or 3, p is a prime number and a Is a positive integer. If a ring R @ Fq x Fpa . Then (Л(R))= (Л(R)) = *( Л(R)) = 3.

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“…This seems to be only slightly less (if at all) than the length of the longest simple chain in the divisor graph or lcm graph on [n] mentioned above. Other papers have looked at tilings of [n] with divisor chains, for example see [3]. This could correspond to asking about the cycle decomposition for permutations in S div (n) or in S lcm (n).…”

Section: Introductionmentioning

confidence: 99%