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Threshold Graphs with an Arbitrary Large Gap Set

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Abstract

An interval in which a given graph has no eigenvalues is called a gap interval. We show that for any real number R greater than \(\frac{1}{2}(-1+\sqrt{2})\), there exist infinitely many threshold graphs with gap interval (0, R). We provide a new recurrence relation for computing the characteristic polynomial of the threshold graphs and based on it, we conclude that the sequence of the least positive (resp. largest negative) eigenvalues of a certain sequence of threshold graphs is convergent. In some particular cases, we compute the limit points.

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Acknowledgements

The authors acknowledge anonymous reviewers for their careful reading and comments that had led to current, improved version of the manuscript. The research of the second author is partially supported by the Science Fund of the Republic of Serbia; grant number 7749676: Spectrally Constrained Signed Graphs with Applications in Coding Theory and Control Theory – SCSG-ctct.

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Authors and Affiliations

  1. Department of Mathematics, Kuwait University, Al-Shadadiyah, Kuwait

    Abdullah Alazemi & Milica Anđelić

  2. Department of Mathematics, College of Graduate Studies, Kuwait University, Al-Shadadiyah, Kuwait

    Haneen Zaidan

Authors
  1. Abdullah Alazemi
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  2. Milica Anđelić
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  3. Haneen Zaidan
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Correspondence to Milica Anđelić.

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Communicated by Fuad Kittaneh.

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Alazemi, A., Anđelić, M. & Zaidan, H. Threshold Graphs with an Arbitrary Large Gap Set. Bull. Malays. Math. Sci. Soc. 47, 88 (2024). https://doi.org/10.1007/s40840-024-01680-w

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  • DOI: https://doi.org/10.1007/s40840-024-01680-w

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