For nontrivial trees, 1/4 H(T) < μ*(T) < 3/2 H(T) with both constants sharp; the TxGraffiti conjecture is false, with F4 and the subdivided star as smallest counterexamples.
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Sharp bounds between the saturation number and the harmonic index
For nontrivial trees, 1/4 H(T) < μ*(T) < 3/2 H(T) with both constants sharp; the TxGraffiti conjecture is false, with F4 and the subdivided star as smallest counterexamples.