Abstract
The harmonic index H ( G ) of a graph G is ∑ u v ∈ E ( G ) 2 d ( u ) + d ( v ) , where d ( v ) is the degree of v ∈ V ( G ) . We show that H ( L ( T ) ) > n 4 for any tree T of order n ≥ 3 , which confirm the validity of a conjecture proposed by Zhang and Wu. In addition, the same lower bound holds for a unicyclic graph G of order n . Two relevant conjectures are proposed as well.
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