The saturation number is not bounded by the harmonic index (opens in new tab)

TxGraffiti conjectured in 2023 that every nontrivial connected graph $G$ satisfies $\mu^*(G) \le H(G)$, where $\mu^*(G)$ is the saturation number and $H(G)$ is the harmonic index. The conjecture is false: the friendship graph $F_4$ satisfies $\mu^*(F_4) = 4 > 18/5 = H(F_4)$, and an exhaustive enumeration confirms that nine vertices is the smallest order admitting a counterexample. A generalized windmill family shows that the ratio $\mu^*/H$ ca...