Hitting Time Index for Broom Graphs

Thehitting time index HT(G) is a recently introduced topological descriptor based on expected hitting times of a random walk on a graph. In this paper, we derive a closed-form formula for HT(G) for broom graphs Bn,d that holds for all parameters 2≤d≤n−1, HT(Bn,d)=S1+S2+S3+(n−d)∑i=1d−1max{A(i),B(i)}, where S1,S2,S3,A(i),B(i) are explicitly defined. For d≥2 and n≥4d−8 we derive a simpler cubic polynomial formula in n, HT(Bn,d)=n3+adn2+bdn+cd, with explicitly given coefficients ad,bd,cd depending only on d. We also consider quartic polynomial formulas for special cases.