Geodesic
Enumeration of labeled nonplanar pentacyclic blocks
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 4, Tome 193 (2021), pp. 28-32

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An planar graph is a graph that can be drawn on a plane without intersecting edges. A pentacyclic graph is a connected graph with $n$ vertices and $n + 4$ edges. We obtain an explicit formula for the number of labeled nonplanar pentacyclic blocks with a given number of vertices and found the corresponding asymptotics for the number of such graphs with a large number of vertices. We prove that under the uniform probability distribution, the probability that the labeled pentacyclic block is a nonplanar graph is asymptotically equal to $80/539$.
Keywords: enumeration, labeled graph, block, planar graph, asymptotics, probability. 
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     author = {V. A. Voblyi},
     title = {Enumeration of labeled nonplanar pentacyclic blocks},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {28--32},
     publisher = {mathdoc},
     volume = {193},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2021_193_a4/}
}
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V. A. Voblyi. Enumeration of labeled nonplanar pentacyclic blocks. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh spring mathematical school “Modern methods of the theory of boundary-value problems. Pontryagin readings – XXX”. Voronezh, May 3-9, 2019. Part 4, Tome 193 (2021), pp. 28-32. http://geodesic.mathdoc.fr/item/INTO_2021_193_a4/