On alternating paths and the smallest positive eigenvalue of trees
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.
KeywordsTree Alternating path Smallest positive eigenvalue Inverse of a tree
Mathematics Subject Classification05C30 05C05 05C35 92E10
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