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Paper IPM / M / 15167  


Abstract:  
Let k ≥ 1 and n_{1},…,n_{k} ≥ 1 be some integers. Let S(n_{1},…,n_{k}) be the tree T such that T has a vertex v of degree k and T\v is the disjoint union of the paths P_{n1},…,P_{nk}, that is T\v ≅ P_{n1}∪…∪P_{nk} such that every neighbor of v in T has degree one or two.
The tree S(n_{1},…,n_{k}) 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 n_{1},…,n_{k} ≥ 2, then [(k−1)/(√{k−2})] < λ_{1}(S(n_{1},…,n_{k})) < [(k)/(√{k−1})], where λ_{1}(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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