Upper bound for the diameter of a tree in the quantum graph theory

  • O. P. Boyko South Ukrainian National Pedagogical University named after K. D. Ushynsky
  • O. M. Martynyuk South Ukrainian National Pedagogical University named after K. D. Ushynsky
  • V. M. Pivovarchik South Ukrainian National Pedagogical University named after K. D. Ushynsky
Keywords: spectral problem, eigenvalue, Neumann boundary condition, Kirchhoff's condition, spectrum, tree

Abstract

UDC 519.177

We study two Sturm – Liouville spectral problems on an equilateral tree with continuity and Kirchhoff conditions at internal vertices and Neumann conditions at pendant vertices (first problem) and with Dirichlet conditions at pendant vertices (second problem). The spectrum of each of these problems consists of infinitely many normal (isolated Fredholm) eigenvalues. It is shown that, knowing the asymptotics of the eigenvalues, it is possible to estimate the diameter of a tree from above for each of these problems.

Author Biography

O. M. Martynyuk, South Ukrainian National Pedagogical University named after K. D. Ushynsky

 

 

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Published
04.10.2022
How to Cite
BoykoO. P., MartynyukO. M., and PivovarchikV. M. “Upper Bound for the Diameter of a Tree in the Quantum Graph Theory”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 8, Oct. 2022, pp. 1020 -28, doi:10.37863/umzh.v74i8.7176.
Section
Research articles