Geodesic
An asymptotics for the number of labelled planar tetracyclic and pentacyclic graphs
Prikladnaâ diskretnaâ matematika, no. 1 (2023), pp. 72-79.

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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 $n\to\infty$. 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.
Keywords: labelled graph, planar graph, tetracyclic graph, pentacyclic graph, block, enumeration, asymptotics, probability. 
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     author = {V. A. Voblyi},
     title = {An asymptotics for the number of labelled planar tetracyclic and pentacyclic graphs},
     journal = {Prikladna\^a diskretna\^a matematika},
     pages = {72--79},
     publisher = {mathdoc},
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     year = {2023},
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     url = {http://geodesic.mathdoc.fr/item/PDM_2023_1_a3/}
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V. A. Voblyi. An asymptotics for the number of labelled planar tetracyclic and pentacyclic graphs. Prikladnaâ diskretnaâ matematika, no. 1 (2023), pp. 72-79. http://geodesic.mathdoc.fr/item/PDM_2023_1_a3/