Geodesic
The second Riddel relation and its consequences
Diskretnyj analiz i issledovanie operacij, Tome 26 (2019) no. 1, pp. 20-32.

Voir la notice de l'article provenant de la source Math-Net.Ru

The second Riddell relation relates the generating functions for the number of labeled connected graphs and the number of labeled blocks. We consider the conditions under which this relation is true for a subclass of connected graphs. Under these conditions, the formulas are valid that express the number of graphs from a subclass of labeled connected graphs trough the generating function of their blocks. By way of application, we obtain expressions for the numbers of labeled connected and 2-connected series-parallel graphs. Bibliogr. 22.
Mots-clés : enumeration, labeled graph, connected graph, 2-connected graph, generating function, block-stable class, series-parallel graph. 
@article{DA_2019_26_1_a1,
     author = {V. A. Voblyi},
     title = {The second {Riddel} relation and its consequences},
     journal = {Diskretnyj analiz i issledovanie operacij},
     pages = {20--32},
     publisher = {mathdoc},
     volume = {26},
     number = {1},
     year = {2019},
     language = {ru},
     url = {https://geodesic-test.mathdoc.fr/item/DA_2019_26_1_a1/}
}
TY  - JOUR
AU  - V. A. Voblyi
TI  - The second Riddel relation and its consequences
JO  - Diskretnyj analiz i issledovanie operacij
PY  - 2019
SP  - 20
EP  - 32
VL  - 26
IS  - 1
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/DA_2019_26_1_a1/
LA  - ru
ID  - DA_2019_26_1_a1
ER  - 
%0 Journal Article
%A V. A. Voblyi
%T The second Riddel relation and its consequences
%J Diskretnyj analiz i issledovanie operacij
%D 2019
%P 20-32
%V 26
%N 1
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/DA_2019_26_1_a1/
%G ru
%F DA_2019_26_1_a1
V. A. Voblyi. The second Riddel relation and its consequences. Diskretnyj analiz i issledovanie operacij, Tome 26 (2019) no. 1, pp. 20-32. https://geodesic-test.mathdoc.fr/item/DA_2019_26_1_a1/

[1] V. A. Voblyi, “On enumeration of labelled connected graphs by the number of cutpoints”, Discrete Math. Appl., 18:1 (2008), 57–69 | DOI | DOI | MR | Zbl

[2] V. A. Voblyi, “A formula for the number of labeled connected graphs”, Diskretn. Anal. Issled. Oper., 19:4 (2012), 48–59 (Russian) | Zbl

[3] V. A. Voblyi, “Enumeration of labeled connected graphs with given order and size”, J. Appl. Ind. Math., 10:2 (2016), 302–310 | DOI | MR | Zbl

[4] V. A. Voblyi, A. K. Meleshko, “Enumeration of labeled block-cactus graphs”, J. Appl. Ind. Math., 8:3 (2014), 422–427 | DOI | MR | Zbl

[5] I. P. Goulden, D. M. Jackson, Combinatorial Enumeration, John Wiley Sons, New York, 1983 | MR | Zbl

[6] V. N. Sachkov, An Introduction to Combinatorial Methods of Discrete Mathematics, MTsNMO, M., 2004 (Russian)

[7] F. Harary, E. M. Palmer, Graphical Enumeration, Acad. Press, New York, 1973 | MR | Zbl

[8] Bergeron F., Labelle G., Leroux P., Combinatorial species and tree-like structures, Camb. Univ. Press, Cambridge, 1998, 457 pp. | MR | Zbl

[9] Bodirsky M., Gimenez O., Kang M., Noy M., “Enumeration and limit laws of series-parallel graphs”, Eur. J. Comb., 28:8 (2007), 2091–2105 | DOI | MR | Zbl

[10] Drmota M., Random trees, Springer, Wien–New York, 2009, 458 pp. | MR | Zbl

[11] Ford G. W., Uhlenbeck G. E., “Combinatorial problems in the theory graphs, I”, Proc. Nat. Acad. Sci. USA, 42:3 (1956), 122–128 | DOI | MR | Zbl

[12] McDiarmid C., Scott A., “Random graphs from a block stable class”, Eur. J. Comb., 58 (2016), 96–106 | DOI | MR | Zbl

[13] Labelle G., Leroux P., Ducharme M. G., “Graph weights arising from Mayer's theory cluster integrals”, Sémin. Lothar. Comb., 54 (2007), B54m | MR

[14] Meir A., Moon J. W., “On nodes of degree two in random trees”, Mathematika, 15:2 (1968), 188–192 | DOI | MR | Zbl

[15] Noy M., “Random planar graphs and beyond”, Proc. Int. Congr. Math. (Seoul, Korea, Aug. 13–21, 2014), v. IV, Seoul, 2014, 407–431 | MR

[16] Radhavan S., “Low-connectivity network design on series-parallel graphs”, Networks, 43:3 (2004), 163–176 | DOI | MR

[17] Riddell R. J., Jr., Uhlenbeck G. E., “On the theory of the virial development of the equation of state of monoatomic gases”, J. Chem. Phys., 21:11 (1953), 2056–2064 | DOI | MR

[18] Stemple J. G., Watkins M. E., “On planar geodetic graphs”, J. Comb. Theory, 4:2 (1968), 101–117 | DOI | MR | Zbl

[19] Takamizawa K., Nishezeki T., Saito N., “Linear-time computability of combinatorial problems on series-parallel graphs”, J. ACM, 29:3 (1982), 623–641 | DOI | MR | Zbl

[20] Tazawa S., “Enumeration of labeled 2-connected Euler graphs”, J. Comb. Inform. Syst. Sci., 23:1–4 (1998), 407–414 | MR | Zbl

[21] The on-line encyclopedia of integer sequences, http://oeis.org

[22] Whitney H., “Non-separable and planar graphs”, Trans. Amer. Math. Soc., 34:2 (1932), 339–362 | DOI | MR