Geodesic
Enumeration of labeled thorn graphs
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 49-54

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A thorn graph is a connected graph that becomes smooth after a single removal of end points together with their incident edges. An explicit formula is obtained for the number of labeled thorn graphs with given numbers of vertices and edges, and the corresponding asymptotics is found for the number of such graphs with a large number of vertices. It is proved that with a uniform probability distribution, almost all labeled connected sparse graphs are not thorn graphs.
Keywords: enumeration, labeled graph, thorn graph, asymptotics, probability. 
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     author = {V. A. Voblyi and N. A. Arkhipova},
     title = {Enumeration of labeled thorn graphs},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {49--54},
     publisher = {mathdoc},
     volume = {210},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2022_210_a5/}
}
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V. A. Voblyi; N. A. Arkhipova. Enumeration of labeled thorn graphs. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry, Mechanics, and Differential Equations, Tome 210 (2022), pp. 49-54. http://geodesic.mathdoc.fr/item/INTO_2022_210_a5/