Open Access

# On maximal multiplicity of eigenvalues of finite-dimensional spectral problem on a graph

### 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.

### 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 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},
URL = {http://mfat.imath.kiev.ua/article/?id=1166},
}

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