Abstract
Let T be a tree of order n>6 with μ≠1 as a positive eigenvalue of multiplicity k. Rowlinson [10] obtained a nice bound for k in terms of n, that is k≤n3. It seems slightly imperfect that the bound n3 for k is not tight. Thus we are motivated to give a sharp upper bound for k in terms of n. Applying the theory of star set and star complement, we prove that k≤n−43 if μ2 is an integer at least 2, and the extremal trees T satisfying k=n−43 are characterized completely. As a corollary, we prove that if n≥16, then k≤n−43 for an arbitrary positive eigenvalue μ≠1.
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