arXiv:2611.05649v1 Announce Type: new Abstract: Let $G$ be a finite simple $d$-regular graph, $d\geq 3$, whose second largest adjacency eigenvalue satisfies $\lambda_2(G)\leq (1-\delta)d$, where $0<\delta\leq 1$. We prove that the Poincar\'e constant satisfies $$\gamma(G,\|\cdot\|_X)\lesssim \frac{qCK}{\delta}$$ whenever $X$ is a real Banach space with a $K$-unconditional Schauder basis and cotype $q$ with constant $C$. This constant compares the mean distance over all vertex pairs with the mean distance across edges. For real Banach lattices the bound is $O(qC/\delta)$. The proof estimates where a random walk first leaves each coordinate level set. As a consequence of the bound, if $n$-vertex graphs in a fixed-degree expander family embed with bounded distortion into spaces with bounded $C,K$, then $q$ must grow at least as fast as $\log n$. This order is attained in $\ell_q^n$.
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