Geodesic
Graphs and algebras of symmetric functions
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 174 (2020), pp. 20-36.

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

We describe an algebraic technique for operating with power series whose coefficients are represented by integrals of symmetric functions $f_n$ defined on the Cartesian powers $\Omega^n$ of a set $\Omega$ with a measure $\mu$. Moreover, each of the coefficient functions $f_n$ is obtained by means of a special mapping from graphs with $n$ labeled vertices belonging to a fixed class. This technique has application to equilibrium statistical mechanics and to problems of enumeration of graphs.
Keywords: graph, commutative algebra, symmetric function, invariant measure, generating function, multiplicative functional. 
@article{INTO_2020_174_a2,
     author = {Yu. P. Virchenko and L. P. Danilova},
     title = {Graphs and algebras of symmetric functions},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {20--36},
     publisher = {mathdoc},
     volume = {174},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_174_a2/}
}
TY  - JOUR
AU  - Yu. P. Virchenko
AU  - L. P. Danilova
TI  - Graphs and algebras of symmetric functions
JO  - Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
PY  - 2020
SP  - 20
EP  - 36
VL  - 174
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/INTO_2020_174_a2/
LA  - ru
ID  - INTO_2020_174_a2
ER  - 
%0 Journal Article
%A Yu. P. Virchenko
%A L. P. Danilova
%T Graphs and algebras of symmetric functions
%J Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory
%D 2020
%P 20-36
%V 174
%I mathdoc
%U http://geodesic.mathdoc.fr/item/INTO_2020_174_a2/
%G ru
%F INTO_2020_174_a2
Yu. P. Virchenko; L. P. Danilova. Graphs and algebras of symmetric functions. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Geometry and Mechanics, Tome 174 (2020), pp. 20-36. http://geodesic.mathdoc.fr/item/INTO_2020_174_a2/

[9] Van der Varden B. L., Algebra, Nauka, M., 1979 ; | MR

[10] Virchenko Yu. P., Ostapenko L. P., “Zadachi perechisleniya grafov s pomechennymi vershinami”, Nauch. ved. BelGU. Mat. Fiz., 44:214 (2016), 150–180

[11] Geilikman B. T., Statisticheskaya teoriya fazovykh prevraschenii, GITTL, M., 1954

[12] Maier Dzh., Geppert-Maier M., Statisticheskaya mekhanika, Mir, M., 1980;

[13] Malyshev V. A., Minlos R. A., Gibbsovskie sluchainye polya. Metod klasternykh razlozhenii, Nauka, M., 1985 | MR

[14] Ore O., Teoriya grafov, Nauka, M., 1980;

[15] Ostapenko L. P., Virchenko Yu. P., “Chislo svyaznykh grafov bez vershin sochleneniya”, Materialy mezhdunarodnoi konferentsii «Voronezhskaya zimnyaya matematicheskaya shkola S. G. Kreina—2016», Nauchnaya kniga, Voronezh, 2016, 310–314

[16] Ryuel D., Statisticheskaya mekhanika. Strogie rezultaty, Mir, M., 1971;

[17] Kharrari F., Teoriya grafov, Mir, M., 1973;

[18] Kharrari F., Palmer E., Perechislenie grafov, Mir, M., 1977;

[19] Mayer J., Harrison S. F., “Statistical mechanics of condensing systems. III”, J. Chem. Phys., 6 (1938), 87–100 | DOI

[20] Mayer J., Harrison S. F., “Statistical mechanics of condensing systems. IV”, J. Chem. Phys., 6 (1938), 101–104 | DOI