For each odd integer
n
≥
3
n \geq 3
, we construct a rank-3 graph
Λ
n
\Lambda _n
with involution
γ
n
\gamma _n
whose real
C
∗
C^*
-algebra
C
R
∗
(
Λ
n
,
γ
n
)
C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda _n, \gamma _n)
is stably isomorphic to the exotic Cuntz algebra
E
n
\mathcal E_n
. This construction is optimal, as we prove that a rank-2 graph with involution
(
Λ
,
γ
)
(\Lambda ,\gamma )
can never satisfy
C
R
∗
(
Λ
,
γ
)
∼
M
E
E
n
C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )\sim _{ME} \mathcal E_n
, and Boersema reached the same conclusion for rank-1 graphs (directed graphs) in [Münster J. Math. 10 (2017), pp. 485–521, Corollary 4.3]. Our construction relies on a rank-1 graph with involution
(
Λ
,
γ
)
(\Lambda , \gamma )
whose real
C
∗
C^*
-algebra
C
R
∗
(
Λ
,
γ
)
C^*_{\scriptscriptstyle \mathbb {R}}(\Lambda , \gamma )
is stably isomorphic to the suspension
S
R
S \mathbb {R}
. In the Appendix, we show that the
i
i
-fold suspension
S
i
R
S^i \mathbb {R}
is stably isomorphic to a graph algebra iff
−
2
≤
i
≤
1
-2 \leq i \leq 1
.