Upper Bound for the Diameter of a Tree in the Quantum Graph Theory

We study two Sturm–Liouville spectral problems on an equilateral tree with continuity and Kirchhoff conditions at the internal vertices and Neumann conditions at the pendant vertices (first problem) and with Dirichlet conditions at the pendant vertices (second problem). The spectrum of each of these...

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Bibliographic Details
Published in:Ukrainian mathematical journal 2023, Vol.74 (8), p.1165-1174
Main Authors: Boyko, O. P., Martynyuk, O. M., Pivovarchik, V. M.
Format: Article
Language:English
Subjects:
Algebra
Analysis
Apexes
Applications of Mathematics
Dirichlet problem
Eigenvalues
Geometry
Graph theory
Mathematics
Mathematics and Statistics
Statistics
Upper bounds
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Summary:We study two Sturm–Liouville spectral problems on an equilateral tree with continuity and Kirchhoff conditions at the internal vertices and Neumann conditions at the pendant vertices (first problem) and with Dirichlet conditions at the pendant vertices (second problem). The spectrum of each of these problems consists of infinitely many normal (isolated Fredholm) eigenvalues. It is shown that if we know the asymptotics of eigenvalues, then it is possible to estimate the diameter of a tree from above for each of these problems.
ISSN:0041-5995
1573-9376
DOI:10.1007/s11253-023-02128-3