Geodesic
Enumeration of labeled Eulerian pentacyclic graphs
Prikladnaâ diskretnaâ matematika, no. 4 (2020), pp. 87-92.

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An Euler graph is a connected graph in which all degrees of vertices are even numbers. A pentacyclic graph is a connected graph with $n$ vertices and $n + 4$ edges. We obtain an explicit formula for the number of labeled Euler pentacyclic graphs 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, given a uniform probability distribution, the probability that a labeled pentacyclic Euler graph is a block (cactus) is asymptotically $53/272$ ($63/272$), respectively.
Keywords: labeled graph, Eulerian graph, pentacyclic graph, block, enumeration, asymptotics, probability. 
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V. A. Voblyi. Enumeration of labeled Eulerian pentacyclic graphs. Prikladnaâ diskretnaâ matematika, no. 4 (2020), pp. 87-92. http://geodesic.mathdoc.fr/item/PDM_2020_4_a5/

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