On minimum number of distinct eigenvalues for a Stieltjes string problem on a tree

  • V. M. Pivovarchik South Ukrainian National Pedagogical University named after K. D. Ushynsky



Spectral problems are considered related to small vibrations of a tree of Stieltjes strings. It is shown that the minimum number of distinct eigenvalues of such a problem equals the maximal length (measured in number of point masses) of paths in the tree.




Author Biography

V. M. Pivovarchik, South Ukrainian National Pedagogical University named after K. D. Ushynsky

Вячеслав Миколайович


Ahmadi, Bahman; Alinaghipour, Fatemeh; Cavers, Michael S.; Fallat, Shaun; Meagher, Karen; Nasserasr, Shahla. Minimum number of distinct eigenvalues of graphs. Electron. J. Linear Algebra 26 (2013), 673--691. doi: 10.13001/1081-3810.1679

Barioli, Francesco; Fallat, Shaun M. On two conjectures regarding an inverse eigenvalue problem for acyclic symmetric matrices. Electron. J. Linear Algebra 11 (2004), 41--50. doi: 10.13001/1081-3810.1120

W. Cauer, Die Verwirklichung von Wechselstromwiderst¨anden vorgeschriebenuer Frequenzabh¨angigkeit, Arch.

Electrotechnik, 17, № 4, 355 – 388 (1926).

Cox, Steven J.; Embree, Mark; Hokanson, Jeffrey M. One can hear the composition of a string: experiments with an inverse eigenvalue problem. SIAM Rev. 54 (2012), no. 1, 157--178. doi: 10.1137/080731037

Leal-Duarte, António; Johnson, Charles R. On the minimum number of distinct eigenvalues for a symmetric matrix whose graph is a given tree. Math. Inequal. Appl. 5 (2002), no. 2, 175--180. doi: 10.7153/mia-05-19

Filimonov, Andrey M.; Kurchanov, Pavel F.; Myshkis, Anatoly D. Some unexpected results in the classical problem of vibrations of the string with $n$ beads when $n$ is large. C. R. Acad. Sci. Paris Sér. I Math. 313 (1991), no. 13, 961--965. MR1143454

Guillemin, Ernst A. Synthesis of passive networks. Theory and methods appropriate to the realization and approximation problems. John wiley and Sons, Inc., New York; Chapman and Hall, Ltd., London 1958 {rm xviii}+741 pp. MR0137480

Filimonov, A. M.; Myshkis, A. D. On properties of large wave effect in classical problem of bead string vibration. J. Difference Equ. Appl. 10 (2004), no. 13-15, 1171--1175. doi: 10.1080/10236190410001652757

Gantmacher, F. P.; Krein, M. G. Oscillation matrices and kernels and small vibrations of mechanical systems. Revised edition. Translation based on the 1941 Russian original. Edited and with a preface by Alex Eremenko. AMS Chelsea Publishing, Providence, RI, 2002. viii+310 pp. ISBN: 0-8218-3171-2 doi: 10.1090/chel/345

J. Genin, J. Maybee, Mechanical vibrations tree, J. Math. Anal. Appl., 45, 746 – 763 (1974).

Gladwell, Graham M. L. Inverse problems in vibration. Second edition. Solid Mechanics and its Applications, 119. Kluwer Academic Publishers, Dordrecht, 2004. xvi+457 pp. ISBN: 1-4020-2670-6 MR2102477

Gladwell, Graham M. L. Matrix inverse eigenvalue problems. Dynamical inverse problems: theory and application, 1--28, CISM Courses and Lect., 529, SpringerWienNewYork, Vienna, 2011. doi: 10.1007/978-3-7091-0696-9_1

Hogben, Leslie. Spectral graph theory and the inverse eigenvalue problem of a graph. Electron. J. Linear Algebra 14 (2005), 12--31. doi: 10.13001/1081-3810.1174

Kim, In-Jae; Shader, Bryan L. Classification of trees each of whose associated acyclic matrices with distinct diagonal entries has distinct eigenvalues. Bull. Korean Math. Soc. 45 (2008), no. 1, 95--99. doi: 10.4134/BKMS.2008.45.1.095

V. A. Marchenko, Введение в теорию обратных задач спектрального анализа (Russia)[[Vvedenie v teoriyu obratny`kh zadach spektral`nogo analiza]], Akta, Kharkiv (2005).

Pivovarchik, V. Existence of a tree of Stieltjes strings corresponding to two given spectra. J. Phys. A 42 (2009), no. 37, 375213, 16 pp. doi: 10.1088/1751-8113/42/37/375213

Pivovarchik, Vyacheslav; Rozhenko, Natalia; Tretter, Christiane. Dirichlet-Neumann inverse spectral problem for a star graph of Stieltjes strings. Linear Algebra Appl. 439 (2013), no. 8, 2263--2292. doi: 10.1016/j.laa.2013.07.003

Pivovarchik, Vyacheslav; Tretter, Christiane. Location and multiplicities of eigenvalues for a star graph of Stieltjes strings. J. Difference Equ. Appl. 21 (2015), no. 5, 383--402. doi: 10.1080/10236198.2014.992425

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

How to Cite
PivovarchikV. M. “On Minimum Number of Distinct Eigenvalues for a Stieltjes String Problem on a Tree”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 72, no. 1, Jan. 2020, pp. 135-41, http://umj.imath.kiev.ua/index.php/umj/article/view/959.
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