Abstract
The harmonic index of graph [Formula: see text] is the value [Formula: see text], where [Formula: see text] refers to the degree of [Formula: see text]. Zhong [The harmonic index for graphs, Appl. Math. Lett. 25 (2012) 561–566] proved that [Formula: see text] for any tree [Formula: see text] of order [Formula: see text]. As a results of Ali, Raza and Bhatti [Some vertex-degree-based topological indices of cacti, Ars Combin. 144 (2019) 195–206], it can be shown that [Formula: see text] for any tree [Formula: see text] of order [Formula: see text] with [Formula: see text] leaves, where [Formula: see text]. In this paper, we generalize this lower bound for all cactus graphs. We present a lower bound for the harmonic index of cactus graphs [Formula: see text] in terms of the order, the number of leaves and the number of cycles, and characterize all cactus graphs achieving equality for the given bound.
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