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Developments on spectral characterizations of graphs

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  • van Dam, E.R.

    (Tilburg University, School of Economics and Management)

  • Haemers, W.H.

    (Tilburg University, School of Economics and Management)

Abstract

In [E.R. van Dam, W.H. Haemers, Which graphs are determined by their spectrum? Linear Algebra Appl. 373 (2003), 241–272] we gave a survey of answers to the question of which graphs are determined by the spectrum of some matrix associated to the graph. In particular, the usual adjacency matrix and the Laplacian matrix were addressed. Furthermore, we formulated some research questions on the topic. In the meantime, some of these questions have been (partially) answered. In the present paper we give a survey of these and other developments.
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Suggested Citation

  • van Dam, E.R. & Haemers, W.H., 2009. "Developments on spectral characterizations of graphs," Other publications TiSEM 7b5eb8d4-dfc2-4392-bb3f-4, Tilburg University, School of Economics and Management.
    • Handle: RePEc:tiu:tiutis:7b5eb8d4-dfc2-4392-bb3f-491efeb35c6d
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    References listed on IDEAS

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    1. 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.
    2. van Dam, E.R. & Spence, E., 2003. "Combinatorial Designs with Two Singular Values I. Uniform Multiplicative Designs," Discussion Paper 2003-67, 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. van Dam, E.R. & Spence, E., 2003. "Combinatorial Designs with Two Singular Values II. Partial Geometric Designs," Discussion Paper 2003-94, Tilburg University, Center for Economic Research.
    5. van Dam, E.R. & Haemers, W.H. & Koolen, J.H. & Spence, E., 2005. "Characterizing Distance-Regularity of Graphs by the Spectrum," Discussion Paper 2005-19, Tilburg University, Center for Economic Research.
    6. van Dam, E.R. & Koolen, J.H., 2004. "A New Family of Distance-Regular Graphs with Unbounded Diameter," Discussion Paper 2004-116, Tilburg University, Center for Economic Research.
    7. Bussemaker, F.C. & Haemers, W.H. & Spence, E., 2000. "The search for pseudo orthogonal Latin squares of order six," Other publications TiSEM 860514f0-4ac6-4563-bf5c-b, Tilburg University, School of Economics and Management.
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    Citations

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    Cited by:

    1. 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.
    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. Lei, Xingyu & Wang, Jianfeng, 2022. "Spectral determination of graphs with one positive anti-adjacency eigenvalue," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    4. 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.
    5. Saeree Wananiyakul & Jörn Steuding & Janyarak Tongsomporn, 2022. "How to Distinguish Cospectral Graphs," Mathematics, MDPI, vol. 10(24), pages 1-24, December.
    6. 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).
    7. 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.
    8. Haemers, W.H. & Ramezani, F., 2009. "Graphs Cospectral with Kneser Graphs," Discussion Paper 2009-76, Tilburg University, Center for Economic Research.
    9. Al-Yakoob, Salem & Kanso, Ali & Stevanović, Dragan, 2022. "On Hosoya’s dormants and sprouts," Applied Mathematics and Computation, Elsevier, vol. 430(C).
    10. 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.
    11. 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.
    12. 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.
    13. 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.
    14. 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.
    15. Dalfo, C. & van Dam, E.R. & Fiol, M.A., 2011. "On perturbations of almost distance-regular graphs," Other publications TiSEM 27186838-0516-45bb-8367-6, Tilburg University, School of Economics and Management.
    16. 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.
    17. 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.
    18. 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.
    19. 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.

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