A Decomposition of the Hitting Time Index Using the Kirchhoff Index and an Asymmetry Term

Abstract: We study the hitting time index, a graph invariant defined in terms of expected hitting times of simple random walks on finite connected graphs, through its relation with the Kirchhoff index. Using the commute time identity and Tetali's formula, we decompose the hitting time index into a Kirchhoff-index term and a nonnegative asymmetry term expressed by degree-weighted effective resistance sums. We derive lower and upper bounds for the asymmetry term, together with their equality conditions, and obtain a lower bound for the hitting time index in terms of the range of the degree-weighted resistance sums. For trees, we obtain lower and upper bounds for the asymmetry term, with the upper bound expressed in terms of the Wiener index, and characterize the equality case in the lower bound. In particular, we show that among all trees of fixed order, the hitting time index is minimized uniquely by the star graph. Finally, we apply the decomposition to complete bipartite graphs, paths, and graphs obtained by conjoining complete graphs, and recover several known hitting-time formulas.