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


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




G. Berkolaiko, P. Kuchment, Introduction to quantum graphs, Amer. Math. Soc., Providence, R.I. (2013), https://doi.org/10.1090/surv/186 DOI: https://doi.org/10.1090/surv/186

F. Barioli, S. Fallat, On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices, Electron. J. Linear Algebra, 11, 41 – 50 (2004), https://doi.org/10.13001/1081-3810.1120 DOI: https://doi.org/10.13001/1081-3810.1120

A. Leal Duarte, C. R. Johnson, On the minimum number of distinct eigenvalues for a symmetric matrix whose graph is a given tree, Math. Inequal. Appl., 5, № 2, 175 – 180 (2002), https://doi.org/10.7153/mia-05-19 DOI: https://doi.org/10.7153/mia-05-19

V. M. Pivovarchik, Pro minimal'nu kil'kist' riznih vlasnih znachen' v zadachi na derevi zi stil't'esivs'kih strun, Ukr. mat. zhurn., 72, № 1, 135 – 141 (2020).

Fan R. K. Chung, Spectral graph theory, Amer. Math. Soc., Providence, R.I. (1997).

C. Cattaneo, The spectrum of the continuous Laplacian on a graph, Monatsh. Math., 124, № 3, 215 – 235 (1997), https://doi.org/10.1007/BF01298245 DOI: https://doi.org/10.1007/BF01298245

P. Exner, A duality between Schrodinger operators on graphs and certain Jacobi matrices, Ann. Inst. H. Poincare A, 66, 359 – 371 (1997).

J. Friedman, J.-P. Tillich, Wave equations for graphs and the edge-based Laplacian, Pacific J. Math., 216, № 2, 229 – 266 (2004), https://doi.org/10.2140/pjm.2004.216.229 DOI: https://doi.org/10.2140/pjm.2004.216.229

R. Carlson, V. Pivovarchik, Spectral asymptotics for quantum graphs with equal edge lengths, J. Phys. A, 41, Article 145202 (2008), https://doi.org/10.1088/1751-8113/41/14/145202 DOI: https://doi.org/10.1088/1751-8113/41/14/145202

A. Chernyshenko, V. Pivovarchik, Recovering the shape of a quantum graph, Int. Equat. Oper. Theory, 92, Article 23 (2020), https://doi.org/10.1007/s00020-020-02581-w DOI: https://doi.org/10.1007/s00020-020-02581-w

A. Chernyshenko, V. Pivovarchik, Cospectral graphs (2022); arXiv:2112.14235 [math-ph] 23 Mar 22.

M. Moller, V. Pivovarchik, Direct and inverse finite-dimensional spectral problems on graphs/, Operator Theory: Adv. and Appl., 283, Birkhauser/Springer (2020); https://www.springer.com/gp/book/9783030604837 DOI: https://doi.org/10.1007/978-3-030-60484-4

YU. V. Pokornyj, O. M. Penkin, V. L. Pryadiev, A. V. Borovskih, K. P. Lazarev, S. A. SHabrov, Differencial'nye uravneniya na geometricheskih grafah, Fizmatlit, Moskva (2005).

M.-E. Pistol, Generating isospectral but not isomorphic quantum graphs; arXiv: 2104.12885 [math. SP] 19 Sep 21.

V. Pivovarchik, On multiplicities of eigenvalues of a boundary value problem on a snowflake graph, Linear Algebra, Appl., 571, 78 – 91 (2019), https://doi.org/10.1016/j.laa.2019.02.012 DOI: https://doi.org/10.1016/j.laa.2019.02.012

How to Cite
Boyko, O. P., O. M. Martynyuk, and V. M. Pivovarchik. “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.
Research articles