Geodesic
Asymptotical enumeration of some abeled geodetic graphs
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Geometry, and Combinatorics, Tome 215 (2022), pp. 58-67.

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We asymptotically enumerate labeled geodetic $k$-cyclic cacti and obtain asymptotics for the numbers of labeled connected geodetic unicyclic, bicyclic, and tricyclic $n$-vertex graphs. We prove that under the uniform probability distribution, the probabilities that a random labeled connected unicyclic, bicyclic, or tricyclic graph is a geodetic graph are asymptotically equal to $1/2$, $3/20$, and $1/30$, respectively. In addition, we prove that almost all labeled connected geodetic tricyclic graphs are cacti.
Keywords: enumeration, labeled graph, geodetic graph, $k$-cyclic graph, asymptotics, random graph.
Mots-clés : cactus 
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V. A. Voblyi. Asymptotical enumeration of some abeled geodetic graphs. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Algebra, Geometry, and Combinatorics, Tome 215 (2022), pp. 58-67. http://geodesic.mathdoc.fr/item/INTO_2022_215_a5/

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