IDEAS home Printed from https://ideas.repec.org/a/spr/indpam/v45y2014i4d10.1007_s13226-014-0079-2.html
   My bibliography  Save this article

Main Q-eigenvalues and generalized Q-cospectrality of graphs

Author

Listed:
  • Tianyi Bu

    (Harbin Engineering University)

  • Lizhu Sun

    (Harbin Engineering University)

  • Wenzhe Wang

    (Harbin Engineering University)

  • Jiang Zhou

    (Harbin Engineering University
    Harbin Engineering University)

Abstract

Let Q G denote the signless Laplacian matrix of a graph G. An eigenvalue μ of Q G is said to be a main Q-eigenvalue of G if μ has an eigenvector which is not orthogonal to an all-ones vector e. We give some basic properties of main Q-eigenvalues. For a graph G of order n, G is called Q-controllable if G has n distinct main Q-eigenvalues. We show that a graph H is generalized Q-cospectral with a Q-controllable G if and only if H is Q-controllable and there exists a unique rational orthogonal matrix R such that R e = e, Q H = R ⊤ Q G R.

Suggested Citation

  • Tianyi Bu & Lizhu Sun & Wenzhe Wang & Jiang Zhou, 2014. "Main Q-eigenvalues and generalized Q-cospectrality of graphs," Indian Journal of Pure and Applied Mathematics, Springer, vol. 45(4), pages 531-538, August.
    • Handle: RePEc:spr:indpam:v:45:y:2014:i:4:d:10.1007_s13226-014-0079-2
    DOI: 10.1007/s13226-014-0079-2
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s13226-014-0079-2
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s13226-014-0079-2?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. van Dam, E.R. & Haemers, W.H., 2007. "Developments on Spectral Characterizations of Graphs," Discussion Paper 2007-33, Tilburg University, Center for Economic Research.
    2. van Dam, E.R. & Haemers, W.H. & Koolen, J.H., 2006. "Cospectral Graphs and the Generalized Adjacency Matrix," Discussion Paper 2006-31, Tilburg University, Center for Economic Research.
    3. van Dam, E.R. & Haemers, W.H., 2002. "Which Graphs are Determined by their Spectrum?," Discussion Paper 2002-66, Tilburg University, Center for Economic Research.
    4. Abiad, A. & Haemers, W.H., 2012. "Cospectral Graphs and Regular Orthogonal Matrices of Level 2," Discussion Paper 2012-042, Tilburg University, Center for Economic Research.
    5. Abiad, A. & Haemers, W.H., 2012. "Cospectral Graphs and Regular Orthogonal Matrices of Level 2," Other publications TiSEM c656ae2c-dc83-44be-906c-b, Tilburg University, School of Economics and Management.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Saeree Wananiyakul & Jörn Steuding & Janyarak Tongsomporn, 2022. "How to Distinguish Cospectral Graphs," Mathematics, MDPI, vol. 10(24), pages 1-24, December.
    2. Xiaoyun Yang & Ligong Wang, 2020. "Laplacian Spectral Characterization of (Broken) Dandelion Graphs," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(3), pages 915-933, September.
    3. B. R. Rakshith, 2022. "Signless Laplacian spectral characterization of some disjoint union of graphs," Indian Journal of Pure and Applied Mathematics, Springer, vol. 53(1), pages 233-245, March.
    4. Haemers, W.H. & Ramezani, F., 2009. "Graphs Cospectral with Kneser Graphs," Discussion Paper 2009-76, Tilburg University, Center for Economic Research.
    5. van Dam, E.R. & Haemers, W.H., 2010. "An Odd Characterization of the Generalized Odd Graphs," Other publications TiSEM 2478f418-ae83-4ac3-8742-2, Tilburg University, School of Economics and Management.
    6. van Dam, E.R. & Haemers, W.H., 2007. "Developments on Spectral Characterizations of Graphs," Discussion Paper 2007-33, Tilburg University, Center for Economic Research.
    7. van Dam, E.R. & Haemers, W.H., 2010. "An Odd Characterization of the Generalized Odd Graphs," Discussion Paper 2010-47, Tilburg University, Center for Economic Research.
    8. Al-Yakoob, Salem & Kanso, Ali & Stevanović, Dragan, 2022. "On Hosoya’s dormants and sprouts," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    9. van Dam, E.R. & Omidi, G.R., 2011. "Graphs whose normalized laplacian has three eigenvalues," Other publications TiSEM d3b7fa76-22b5-4a9a-8706-a, Tilburg University, School of Economics and Management.
    10. Xue, Jie & Liu, Shuting & Shu, Jinlong, 2018. "The complements of path and cycle are determined by their distance (signless) Laplacian spectra," Applied Mathematics and Computation, Elsevier, vol. 328(C), pages 137-143.
    11. Yuan, Bo-Jun & Wang, Yi & Xu, Jing, 2020. "Characterizing the mixed graphs with exactly one positive eigenvalue and its application to mixed graphs determined by their H-spectra," Applied Mathematics and Computation, Elsevier, vol. 380(C).
    12. Aouchiche, Mustapha & Hansen, Pierre, 2018. "Cospectrality of graphs with respect to distance matrices," Applied Mathematics and Computation, Elsevier, vol. 325(C), pages 309-321.
    13. Lei, Xingyu & Wang, Jianfeng, 2022. "Spectral determination of graphs with one positive anti-adjacency eigenvalue," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    14. Fei Wen & Qiongxiang Huang & Xueyi Huang & Fenjin Liu, 2015. "The spectral characterization of wind-wheel graphs," Indian Journal of Pure and Applied Mathematics, Springer, vol. 46(5), pages 613-631, October.
    15. van Dam, E.R. & Haemers, W.H., 2007. "Developments on Spectral Characterizations of Graphs," Other publications TiSEM 3827c785-7b51-4bcd-aea3-f, Tilburg University, School of Economics and Management.
    16. Maiorino, Enrico & Rizzi, Antonello & Sadeghian, Alireza & Giuliani, Alessandro, 2017. "Spectral reconstruction of protein contact networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 471(C), pages 804-817.
    17. Fei Wen & Qiongxiang Huang & Xueyi Huang & Fenjin Liu, 2018. "On the Laplacian spectral characterization of Π-shape trees," Indian Journal of Pure and Applied Mathematics, Springer, vol. 49(3), pages 397-411, September.
    18. Haemers, W.H. & Ramezani, F., 2009. "Graphs Cospectral with Kneser Graphs," Other publications TiSEM 386fd2ad-65f2-42b6-9dfc-1, Tilburg University, School of Economics and Management.
    19. Xihe Li & Ligong Wang & Shangyuan Zhang, 2018. "The Signless Laplacian Spectral Radius of Some Strongly Connected Digraphs," Indian Journal of Pure and Applied Mathematics, Springer, vol. 49(1), pages 113-127, March.
    20. Lizhu Sun & Wenzhe Wang & Jiang Zhou & Changjiang Bu, 2015. "Laplacian spectral characterization of some graph join," Indian Journal of Pure and Applied Mathematics, Springer, vol. 46(3), pages 279-286, June.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:indpam:v:45:y:2014:i:4:d:10.1007_s13226-014-0079-2. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.