Exact absorption time for stars under birth-death updating

Prove that for birth-death updating on an undirected star graph with N vertices (n leaves plus one center, so N = n + 1), starting from maximum diversity where all N individuals represent distinct types, the exact expected absorption time equals T_N = n^3 − n^2 + n·H_n, where H_n = ∑_{j=1}^n (1/j) is the nth harmonic number.

Background

The paper establishes that the expected absorption time (time to reach homogeneity) for undirected star graphs under birth-death (bd) updating scales as Θ(N^3). The authors further propose a precise closed-form expression for the exact expected time, relating it to the number of leaves n and the harmonic number H_n.

This conjecture refines the asymptotic result by specifying the exact dependence on n, and proving it would clarify the detailed dynamics of neutral drift on highly heterogeneous population structures such as stars.