The least doubling constant of a path graph

We study the least doubling constant $C_G$ among all possible doubling measures defined on a path graph $G$. We consider both finite and infinite cases and show that, if $G=\mathbb Z$, $C_{\mathbb Z}=3$, while for $G=L_n$, the path graph with $n$ vertices, one has $1+2\cos(\frac{\pi}{n+1})\leq C_{L_n}<3$, with equality on the lower bound if and only if $n\le8$. Moreover, we analyze the structure of doubling minimizers on $L_n$ and $\mathbb Z$, those measures whose doubling constant is the smallest possible.