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Closeness spectra and structural uniqueness of special graph classes

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  • Farhad, Qaisar
  • Hussain, Mumtaz
  • Xu, Shou-Jun

Abstract

For a graph G with vertex set V(G) and edge set E(G). Let d(u,v) be the distance between vertices u and v. The closeness matrix of a graph G is a symmetric matrix, where each entry cG(u,v) is defined as cG(u,v)=2−d(u,v) for u≠v, and cG(u,v)=0, if u=v. In the present study, we investigate the closeness spectra of connected graphs and inquire whether specific graph classes may be distinguished by their closeness eigenvalues. We inspect the tree T3,n−3, the path Pn, the complete graph Kn, and the join graph (P1∪P3)∨Kn−4‾. Applying careful spectral computation, we demonstrate that such graphs are uniquely specified by their closeness spectra. Additionally, we confirm that the tree T3,n−3 achieves the second-smallest closeness eigenvalue among trees of diameter 3. Our findings emphasize the importance of closeness matrices in spectral graph theory and lead to more clarity of the relationship between the structure of graphs and its spectrum.

Suggested Citation

  • Farhad, Qaisar & Hussain, Mumtaz & Xu, Shou-Jun, 2026. "Closeness spectra and structural uniqueness of special graph classes," Applied Mathematics and Computation, Elsevier, vol. 509(C).
    • Handle: RePEc:eee:apmaco:v:509:y:2026:i:c:s0096300325003923
    DOI: 10.1016/j.amc.2025.129666
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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. Xue, Jie & Liu, Ruifang & Shu, Jinlong, 2021. "On graphs whose third largest distance eigenvalue dose not exceed −1," Applied Mathematics and Computation, Elsevier, vol. 402(C).
    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. Das, Kinkar Ch. & Liu, Muhuo, 2017. "Kite graphs determined by their spectra," Applied Mathematics and Computation, Elsevier, vol. 297(C), pages 74-78.
    5. Dangalchev, Chavdar, 2006. "Residual closeness in networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 365(2), pages 556-564.
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