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Paper   IPM / M / 15167
School of Mathematics
  Title:   On the eigenvalues and spectral radius of starlike trees
  Author(s):  Mohammad Reza Oboudi
  Status:   Published
  Journal: Aequat. Math.
  Year:  2018
  Pages:   DOI: 10.1007/s00010-017-0533-4
  Supported by:  IPM
Let k ≥ 1 and n1,…,nk ≥ 1 be some integers. Let S(n1,…,nk) be the tree T such that T has a vertex v of degree k and T\v is the disjoint union of the paths Pn1,…,Pnk, that is T\vPn1∪…∪Pnk such that every neighbor of v in T has degree one or two. The tree S(n1,…,nk) is called starlike tree, a tree with exactly one vertex of degree greater than two, if k ≥ 3. In this paper we obtain the eigenvalues of starlike trees. We obtain some bounds for the largest eigenvalue ( for the spectral radius) of starlike trees. In particular we prove that if k ≥ 4 and n1,…,nk ≥ 2, then [(k−1)/(√{k−2})] < λ1(S(n1,…,nk)) < [(k)/(√{k−1})], where λ1(T) is the largest eigenvalue of T. Finally we characterize all starlike trees that all of whose eigenvalues are in the interval (−2,2).

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