Abstract
Recurrence relations of the second order on the edges of a metric connected
graph together with boundary and matching conditions at the
vertices generate a spectral problem for a self-adjoint finite-dimensional
operator. This spectral problem describes small transverse vibrations of a
graph of Stieltjes strings. It is shown that if the
graph is cyclically connected and the number of masses on each edge is not less
than 3 then the maximal multiplicity of an
eigenvalue is $\mu+1$ where $\mu$ is the cyclomatic number of the graph. If
the graph is not cyclically connected and each edge of it bears at least one
point mass then the
maximal multiplicity of an eigenvalue is expressed via $\mu$, the number of
edges and the number
of interior vertices in the tree obtained by
contracting all the cycles of the graph into vertices.
Key words: Tree, cycle, eigenvalue.
Full Text
Article Information
| Title | On maximal multiplicity of eigenvalues of finite-dimensional spectral
problem on a graph |
| Source | Methods Funct. Anal. Topology, Vol. 25 (2019), no. 2, 104-117 |
| MathSciNet |
MR3978675 |
| Milestones | Received 03/06/2018 |
| Copyright | The Author(s) 2019 (CC BY-SA) |
Authors Information
Olga Boiko
South-Ukrainian National Pedagogical University, 26 Staroportofrankovskaya, Odesa, Ukraine
Olga Martynyuk
South-Ukrainian National Pedagogical University, 26 Staroportofrankovskaya, Odesa, Ukraine
Vyacheslav Pivovarchik
South-Ukrainian National Pedagogical University, 26 Staroportofrankovskaya, Odesa, Ukraine
Citation Example
Olga Boiko, Olga Martynyuk, and Vyacheslav Pivovarchik, On maximal multiplicity of eigenvalues of finite-dimensional spectral
problem on a graph, Methods Funct. Anal. Topology 25
(2019), no. 2, 104-117.
BibTex
@article {MFAT1166,
AUTHOR = {Olga Boiko and Olga Martynyuk and Vyacheslav Pivovarchik},
TITLE = {On maximal multiplicity of eigenvalues of finite-dimensional spectral
problem on a graph},
JOURNAL = {Methods Funct. Anal. Topology},
FJOURNAL = {Methods of Functional Analysis and Topology},
VOLUME = {25},
YEAR = {2019},
NUMBER = {2},
PAGES = {104-117},
ISSN = {1029-3531},
MRNUMBER = {MR3978675},
URL = {http://mfat.imath.kiev.ua/article/?id=1166},
}
