Abstract
In this article, we undertake the problem of finding the first four trees on a fixed number of vertices with the maximum smallest positive eigenvalue. Let \({\mathcal {T}}_{n,d}\) denote the class of trees on n vertices with diameter d. First, we obtain the bounds on the smallest positive eigenvalue of trees in \({\mathcal {T}}_{n,d}\) for \(d =2,3,4\) and then upper bounds on the smallest positive eigenvalue of trees are obtained in general class of all trees on n vertices. Finally, the first four trees on n vertices with the maximum, second maximum, third maximum and fourth maximum smallest positive eigenvalue are characterized.
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Funding was provided by Council of Scientific and Industrial Research, India (Grant No. 09/1059(0007)/2004-EMR-I).
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Communicated by Kolja Knauer.
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Rani, S., Barik, S. Upper Bounds on the Smallest Positive Eigenvalue of Trees. Ann. Comb. 27, 19–29 (2023). https://doi.org/10.1007/s00026-022-00619-x
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DOI: https://doi.org/10.1007/s00026-022-00619-x