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The General Extended Adjacency Eigenvalues of Chain Graphs

Author

Listed:
  • Bilal Ahmad Rather

    (Department of Mathematical Sciences, College of Science, United Arab Emirates University, Al Ain 15551, United Arab Emirates)

  • Hilal A. Ganie

    (Department of School Education, Jammu and Kashmir Government, Srinagar 193404, Kashmir, India)

  • Kinkar Chandra Das

    (Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea)

  • Yilun Shang

    (Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK)

Abstract

In this article, we discuss the spectral properties of the general extended adjacency matrix for chain graphs. In particular, we discuss the eigenvalues of the general extended adjacency matrix of the chain graphs and obtain its general extended adjacency inertia. We obtain bounds for the largest and the smallest general extended adjacency eigenvalues and characterize the extremal graphs. We also obtain a lower bound for the spread of the general extended adjacency matrix. We characterize chain graphs with all the general extended adjacency eigenvalues being simple and chain graphs that are non-singular under the general extended adjacency matrix. Further, we determine the explicit formula for the determinant and the trace of the square of the general extended adjacency matrix of chain graphs. Finally, we discuss the energy of the general extended adjacency matrix and obtain some bounds for it. We characterize the extremal chain graphs attaining these bounds.

Suggested Citation

  • Bilal Ahmad Rather & Hilal A. Ganie & Kinkar Chandra Das & Yilun Shang, 2024. "The General Extended Adjacency Eigenvalues of Chain Graphs," Mathematics, MDPI, vol. 12(2), pages 1-16, January.
    • Handle: RePEc:gam:jmathe:v:12:y:2024:i:2:p:192-:d:1314352
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    References listed on IDEAS

    as
    1. Milovanović, Igor & Milovanović, Emina & Gutman, Ivan, 2016. "Upper bounds for some graph energies," Applied Mathematics and Computation, Elsevier, vol. 289(C), pages 435-443.
    2. Rather, Bilal A. & Ganie, Hilal A. & Shang, Yilun, 2023. "Distance Laplacian spectral ordering of sun type graphs," Applied Mathematics and Computation, Elsevier, vol. 445(C).
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    Cited by:

    1. Bilal A. Rather & Hilal A. Ganie & Yilun Shang, 2024. "On the Signless Laplacian ABC -Spectral Properties of a Graph," Mathematics, MDPI, vol. 12(15), pages 1-23, July.
    2. Zhuo-Heng He & Jie Tian & Shao-Wen Yu, 2024. "A System of Four Generalized Sylvester Matrix Equations over the Quaternion Algebra," Mathematics, MDPI, vol. 12(15), pages 1-26, July.

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