Harmonic index upper bounds the minimum maximal matching size

Prove that for every nontrivial connected finite simple undirected graph G, the minimum cardinality of a maximal matching μ*(G) satisfies μ*(G) ≤ H(G), where H(G) is the harmonic index; moreover, demonstrate that this bound is sharp.

Background

The harmonic index H(G) is a degree-based topological index introduced by Fajtlowicz and studied in chemical graph theory, defined as H(G) = Σ_{uv∈E(G)} 2/(d(u)+d(v)). The parameter μ*(G) is the minimum cardinality of a maximal matching (also known as the saturation number).

The conjecture proposes a direct inequality bridging a continuous, degree-sensitive invariant with a discrete saturation parameter, a link not commonly observed in the literature. Empirical evidence across hundreds of graphs supports both the inequality and its sharpness on structured families.