The continuity of biased random walk's spectral radius on free product graphs

R. Lyons, R. Pemantle and Y. Peres (Ann. Probab. 24 (4), 1996, 1993–2006) conjectured that for a Cayley graph $G$ with a growth rate $\mathrm{g}\mathrm{r}(G)>1$, the speed of a biased random walk exists and is positive for the biased parameter $\lambda \in (1,\mathrm{g}\mathrm{r}(G))$. And Gábor Pete (Probability and geometry on groups, Chaper 9, 2024) sheds light on the intricate relationship between the spectral radius of the graph and the speed of the biased random walk. Here, we focus on an example of a Cayley graph, a free product of complete graphs. In this paper, we establish the continuity of the spectral radius of biased random walks with respect to the bias parameter in this class of Cayley graphs. Our method relies on the Kesten-Cheeger-Dodziuk-Mohar theorem and the analysis of generating functions.