Harary index, binding number and toughness of graphs - Mingqiang An (Kuwait Journal of Science)
Given a connected (molecular) graph 𝐺, its Harary index 𝐻(𝐺) is defined by the reciprocal sum of the distances between all unordered pairs of vertices of 𝐺. Lower bounds on the graph vulnerability parameters binding number and toughness have been often used to determine that the graph has a certain property. Recently, Yatauro [Discrete Appl. Math. 338 (2023) 56-68] gave a sharp upper bound of 𝑊 (𝐺) (Wiener index) to ensure that 𝐺 has a determined lower bound (greater than or equal to 1) for the binding number or toughness. In this paper, we supply an upper bound on 𝐻(𝐺) to ensure that 𝐺 has a determined lower bound (between 0 and 1) for the binding number or toughness and show that these bounds are best possible in some sense.