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Spectral determination of graphs with one positive anti-adjacency eigenvalue

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  • Lei, Xingyu
  • Wang, Jianfeng

Abstract

The anti-adjacency matrix (or eccentricity matrix) of a graph is obtained from its distance matrix by retaining for each row and each column only the largest distances. This matrix can be viewed as the opposite of the adjacency matrix, which is, on the contrary, obtained from the distance matrix of a graph by keeping for each row and each column only the distances being 1. In this paper, we prove that the graphs with exactly one positive anti-adjacency eigenvalue are determined by the anti-adjacency spectra. As corollaries, the well-known (generalized) friendship graphs and windmill graphs are shown to be determined by their anti-adjacency spectra.

Suggested Citation

  • Lei, Xingyu & Wang, Jianfeng, 2022. "Spectral determination of graphs with one positive anti-adjacency eigenvalue," Applied Mathematics and Computation, Elsevier, vol. 422(C).
    • Handle: RePEc:eee:apmaco:v:422:y:2022:i:c:s0096300322000819
    DOI: 10.1016/j.amc.2022.126995
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    References listed on IDEAS

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    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. Haemers, Willem H. & Sorgun, Sezer & Topcu, Hatice, 2019. "On the spectral characterization of mixed extensions of P 3," Other publications TiSEM d3aad4e5-24f2-499d-a57a-9, Tilburg University, School of Economics and Management.
    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, Aida & Alfaro, Carlos A., 2021. "Enumeration of cospectral and coinvariant graphs," Applied Mathematics and Computation, Elsevier, vol. 408(C).
    5. Topcu, Hatice & Sorgun, Sezer, 2018. "The kite graph is determined by its adjacency spectrum," Applied Mathematics and Computation, Elsevier, vol. 330(C), pages 134-142.
    6. Das, Kinkar Ch. & Liu, Muhuo, 2017. "Kite graphs determined by their spectra," Applied Mathematics and Computation, Elsevier, vol. 297(C), pages 74-78.
    7. 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).
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