This question is from M.I. Ostrovskii (2005): Sobolev spaces on graphs, Quaestiones Mathematicae, 28:4, 501-523
3.2. The Banach-Mazur distances between $S_p(G)$ and $l_p^m$ in terms of Cheeger constants. Theorems 3 and 4 imply estimates from above for the Banach-Mazur distance between $S_p(G)$ and $l_p^m$ of the same dimension in terms of the Cheeger constant. The purpose of this section is to present such estimates. For a connected graph $G$ by $l_p^{\circ}\left(V_G\right)$ we denote the 1-codimensional subspace of $l_p\left(V_G\right)$ given by $\sum_{v \in V(G)} f(v) d_v=0$. It is easy to see that $$ d\left(l_p^{\circ}\left(V_G\right), l_p^m\right) \leq 9, $$ where $m=\left|V_G\right|-1$.