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For each odd integer $n \geq 3$, we construct a rank-3 graph $\Lambda_n$ withinvolution $\gamma_n$ whose real C*-algebra $C^*_\mathbb{R}(\Lambda_n,\gamma_n)$ is stably isomorphic to the exotic Cuntz algebra $\mathcalE_n^\mathbb{R}$. This construction is optimal, as we prove that a rank-2 graphwith involution $(\Lambda,\gamma)$ can never satisfy $C^*_\mathbb{R}(\Lambda,\gamma)\sim_{ME} \mathcal E_n^\mathbb{R}$, and the first author reached thesame conclusion in previous work. Our construction relies on a rank-1 graphwith involution $(\Lambda, \gamma)$ whose real C*-algebra$C^*_\mathbb{R}(\Lambda, \gamma)$ is stably isomorphic to the suspension $ S\mathbb{R}$. In the Appendix, we show that the i-fold suspension $S^i\mathbb{R}$ is stably isomorphic to a graph algebra iff $-2 \leq i \leq 1$.

The Stable Exotic Cuntz Algebras are Higher-Rank Graph Algebras