  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.
