A broom graph is a linear graph with some pendant vertices attached to one of its ends. Using the formula for the commute time of the random walk between two vertices of the graph, which is given in terms of the effective resistance between the vertices, we find closed-form formulas for the hitting time index of some families of broom graphs, extending results found by He et al.

Thehitting time index $\mathrm{HT}\left(G\right)$ 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 $\mathrm{HT}\left(G\right)$ for broom graphs $B_{n,d}$ that holds for all parameters $2\le d\le n-1$, $\mathrm{HT}\left(B_{n,d}\right)=S_1+S_2+S_3+(n-d){\displaystyle \sum _{i=1}^{d-1}}\max \{A\left(i\right),B\left(i\right)\},$ where $S_1,S_2,S_3,A\left(i\right),B\left(i\right)$ are explicitly defined. For $d\ge 2$ and $n\ge 4d-8$ we derive a simpler cubic polynomial formula in n, $\mathrm{HT}\left(B_{n,d}\right)=n^3+a_d{n}^2+b_{d}n+c_d,$ with explicitly given coefficients $a_d,b_d,c_d$ depending only on d. We also consider quartic polynomial formulas for special cases.