Geodesic
Estimate of the First Eigenvalue of the Laplacian on a Graph
Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 885-895.

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

The eigenvalue problem for the Laplacian with Dirichlet boundary conditions on a graph is considered. The main result is an estimate of the first (minimal) eigenvalue. The proof is based on the Schwartz symmetrization of a function on a graph and on its properties.
Keywords: eigenvalue problem, Schwartz symmetrization, Laplacian on a graph. 
@article{MZM_2014_96_6_a7,
     author = {A. T. Diab and P. A. Kuleshov and O. M. Penkin},
     title = {Estimate of the {First} {Eigenvalue} of the {Laplacian} on a {Graph}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {885--895},
     publisher = {mathdoc},
     volume = {96},
     number = {6},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a7/}
}
TY  - JOUR
AU  - A. T. Diab
AU  - P. A. Kuleshov
AU  - O. M. Penkin
TI  - Estimate of the First Eigenvalue of the Laplacian on a Graph
JO  - Matematičeskie zametki
PY  - 2014
SP  - 885
EP  - 895
VL  - 96
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a7/
LA  - ru
ID  - MZM_2014_96_6_a7
ER  - 
%0 Journal Article
%A A. T. Diab
%A P. A. Kuleshov
%A O. M. Penkin
%T Estimate of the First Eigenvalue of the Laplacian on a Graph
%J Matematičeskie zametki
%D 2014
%P 885-895
%V 96
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a7/
%G ru
%F MZM_2014_96_6_a7
A. T. Diab; P. A. Kuleshov; O. M. Penkin. Estimate of the First Eigenvalue of the Laplacian on a Graph. Matematičeskie zametki, Tome 96 (2014) no. 6, pp. 885-895. http://geodesic.mathdoc.fr/item/MZM_2014_96_6_a7/

[1] M. A. Krasnoselskii, Polozhitelnye resheniya operatornykh uravnenii. Glavy nelineinogo analiza, Sovremennye problemy matematiki, Fizmatgiz, M., 1962 | MR | Zbl

[2] M. A. Krasnoselskii, E. A. Lifshits, V. I. Sobolev, Lineinye pozitivnye sistemy. Metod polozhitelnykh operatorov, Fizmatgiz, M., 1985 | MR | Zbl

[3] Yu. V. Pokornyi, O. M. Penkin, V. L. Pryadiev, A. V. Borovskikh, K. P. Lazarev, S. A. Shabrov, Differentsialnye uravneniya na geometricheskikh grafakh, Fizmatlit, M., 2005 | MR

[4] P. Kuchment, “Quantum graphs. I. Some basic structures”, Waves Random Media, 14 (2004), 107–128 | MR

[5] E. Lib, M. Loss, Analiz, Universitetskaya seriya, 1, Nauchnaya kniga, Novosibirsk, 1998 | MR | Zbl

[6] G. Polia, G. Sege, Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatlit, M., 1962 | MR | Zbl

[7] A. V. Komarov, O. M. Penkin, Yu. V. Pokornyi, “O chastotnom spektre mnogomernogo analoga tkanoi membrany”, Dokl. RAN, 390:2 (2003), 151–154 | MR | Zbl

[8] K. Berzh, Teoriya grafov i ee primeneniya, IL, M., 1962 | MR | Zbl