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Abstract
A connected graph with a cyclomatic number k is said to be a k-cyclic graph. We obtain the formula for the number of labelled non-planar pentacyclic graphs with a given number of vertices, and find the asymptotics of the number of labelled connected planar tetracyclic and pentacyclic graphs with n vertices as n → ∞. We prove that under a uniform probability distribution on the set of graphs under consideration, the probability that the labelled tetracyclic graph is planar is asymptotically equal to 1089/1105, and the probability that the labeled pentacyclic graph is planar is asymptotically equal to 1591/1675.
Highlights
We obtain the formula for the number of labelled non-planar pentacyclic graphs with a given number of vertices, and find the asymptotics of the number of labelled connected planar tetracyclic and pentacyclic graphs with n vertices as n → ∞
We prove that under a uniform probability distribution on the set of graphs under consideration, the probability that the labelled tetracyclic graph is planar is asymptotically equal to 1089/1105, and the probability that the labeled pentacyclic graph is planar is asymptotically equal to 1591/1675
 äàííîé ðàáîòå ïîëó÷åíà ôîðìóëà äëÿ ÷èñëà ïîìå÷åííûõ ñâÿçíûõ íåïëàíàðíûõ ïåíòàöèêëè÷åñêèõ ãðàôîâ ñ çàäàííûì ÷èñëîì âåðøèí, à òàêæå íàéäåíà àñèìïòîòèêà
Summary
Ïîëó÷åíà ôîðìóëà äëÿ ÷èñëà ïîìå÷åííûõ íåïëàíàðíûõ ïåíòàöèêëè÷åñêèõ ãðàôîâ ñ çàäàííûì ÷èñëîì âåðøèí, à òàêæå íàéäåíà àñèìïòîòèêà ÷èñëà ïîìå÷åííûõ ñâÿçíûõ ïëàíàðíûõ òåòðàöèêëè÷åñêèõ è ïåíòàöèêëè÷åñêèõ ãðàôîâ ñ n âåðøèíàìè ïðè n → ∞. AN ASYMPTOTICS FOR THE NUMBER OF LABELLED PLANAR TETRACYCLIC AND PENTACYCLIC GRAPHS Àñèìïòîòèêà ÷èñëà ïëàíàðíûõ òåòðàöèêëè÷åñêèõ è ïåíòàöèêëè÷åñêèõ ãðàôîâ Å. Áåíäåð íàø1⁄4ë àñèìïòîòèêó ÷èñëà ïîìå÷åííûõ 2-ñâÿçíûõ ïëàíàðíûõ ãðàôîâ ñ n âåðøèíàìè ïðè n → ∞ [7]. Îäíàêî â îáùåì ñëó÷àå àñèìïòîòèêà ÷èñëà ïîìå÷åííûõ ïëàíàðíûõ n-âåðøèííûõ k-öèêëè÷åñêèõ ãðàôîâ ïðè n → ∞ íåèçâåñòíà.
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