On alternating paths and the smallest positive eigenvalue of trees

Sonu Rani () and Sasmita Barik ()
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Sonu Rani: Indian Institute of Technology, Bhubaneswar
Sasmita Barik: Indian Institute of Technology, Bhubaneswar

Journal of Combinatorial Optimization, 2020, vol. 39, issue 2, No 16, 589-601

Abstract: Abstract In this article, we consider the class of tress on a fixed number of vertices. We consider the problem of finding trees with first four minimum smallest positive eigenvalues. First we obtain the upper and lower bounds on number of alternating paths in a tree. It is shown that the smallest positive eigenvalue of a tree is related to the number of alternating paths in it. With the help of combinatorial arguments, the trees with the maximum, second maximum and third maximum number of alternating paths are derived. Subsequently, the unique trees with the second minimum, third minimum and fourth minimum smallest positive eigenvalue are characterized.

Keywords: Tree; Alternating path; Smallest positive eigenvalue; Inverse of a tree; 05C30; 05C05; 05C35; 92E10 (search for similar items in EconPapers)
Date: 2020
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DOI: 10.1007/s10878-019-00503-0

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