A result related to the largest eigenvalue of a tree

Gurusamy Rengasamy Vijayakumar

Discussiones Mathematicae Graph Theory (2008)

  • Volume: 28, Issue: 3, page 557-561
  • ISSN: 2083-5892

Abstract

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In this note we prove that {0,1,√2,√3,2} is the set of all real numbers l such that the following holds: every tree having an eigenvalue which is larger than l has a subtree whose largest eigenvalue is l.

How to cite

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Gurusamy Rengasamy Vijayakumar. "A result related to the largest eigenvalue of a tree." Discussiones Mathematicae Graph Theory 28.3 (2008): 557-561. <http://eudml.org/doc/270599>.

@article{GurusamyRengasamyVijayakumar2008,
abstract = {In this note we prove that \{0,1,√2,√3,2\} is the set of all real numbers l such that the following holds: every tree having an eigenvalue which is larger than l has a subtree whose largest eigenvalue is l.},
author = {Gurusamy Rengasamy Vijayakumar},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {eigenvalues of a graph; characteristic polynomial},
language = {eng},
number = {3},
pages = {557-561},
title = {A result related to the largest eigenvalue of a tree},
url = {http://eudml.org/doc/270599},
volume = {28},
year = {2008},
}

TY - JOUR
AU - Gurusamy Rengasamy Vijayakumar
TI - A result related to the largest eigenvalue of a tree
JO - Discussiones Mathematicae Graph Theory
PY - 2008
VL - 28
IS - 3
SP - 557
EP - 561
AB - In this note we prove that {0,1,√2,√3,2} is the set of all real numbers l such that the following holds: every tree having an eigenvalue which is larger than l has a subtree whose largest eigenvalue is l.
LA - eng
KW - eigenvalues of a graph; characteristic polynomial
UR - http://eudml.org/doc/270599
ER -

References

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  1. [1] M. Doob, A surprising property of the least eigenvalue of a graph, Linear Algebra and Its Applications 46 (1982) 1-7, doi: 10.1016/0024-3795(82)90021-0. Zbl0503.05044
  2. [2] C. Godsil and G. Royle, Algebraic Graph Theory (Springer, New York, 2001). Zbl0968.05002
  3. [3] P.W.H. Lemmens and J.J. Seidel, Equiangular lines, Journal of Algebra 24 (1973) 494-512, doi: 10.1016/0021-8693(73)90123-3. Zbl0255.50005
  4. [4] L. Lovász, Combinatorial Problems and Exercises (North-Holland Publishing Company, Amsterdam, 1979). 
  5. [5] A.J. Schwenk, Computing the characteristic polynomial of a graph, in: Graphs and Combinatorics, eds. R.A. Bari and F. Harary, Springer-Verlag, Lecture Notes in Math. 406 (1974) 153-172. 
  6. [6] N.M. Singhi and G.R. Vijayakumar, Signed graphs with least eigenvalue < -2, European J. Combin. 13 (1992) 219-220, doi: 10.1016/0195-6698(92)90027-W. Zbl0769.05065
  7. [7] J.H. Smith, Some properties of the spectrum of a graph, in: Combinatorial Structures and their Applications, eds. R. Guy, H. Hanani, N. Sauer and J. Schönheim, Gordon and Breach, New York (1970), 403-406. 
  8. [8] D.B. West, Introduction to Graph Theory, Second edition (Printice Hall, New Jersey, USA, 2001). 

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