Journal of Combinatorial Optimization

, Volume 39, Issue 2, pp 589–601 | Cite as

On alternating paths and the smallest positive eigenvalue of trees

  • Sonu RaniEmail author
  • Sasmita Barik


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.


Tree Alternating path Smallest positive eigenvalue Inverse of a tree 

Mathematics Subject Classification

05C30 05C05 05C35 92E10 



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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Basic SciencesIndian Institute of Technology, BhubaneswarBhubaneswarIndia

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