Extremal connectivity codes in small-degree Ramanujan graphs

Determine whether every d-regular Ramanujan graph with d≥4 satisfies m(H)=2^d, where m(H) is the maximum size of a connectivity code in the graph H.

Background

The paper proves that f(d)=2d for every d≥4 by constructing infinitely many d-regular bipartite graphs with extremal linear connectivity codes. These examples arise as cyclic lifts of K_{d,d} and are explicitly noted to be far from Ramanujan graphs.

The unresolved question asks whether the same extremal value holds for every d-regular Ramanujan graph in degrees d≥4. This would extend the known result from the paper's particular construction to a substantially broader and more spectrally constrained class of graphs. The authors state that their construction does not address this question.

References

Alon asks in whether $m(H)=2{d}$ for every $d$-regular Ramanujan graph with $d\ge 4$. Our examples are cyclic lifts of $K_{d,d}$ and are far from Ramanujan, so this remains open.

— Alon's Question on Connectivity Graph-Codes: $f(d)=2^d$ for Every $d\geq 4$  (2609.02953 - Tian, 2 Sep 2026) in Section 6, Concluding remarks, item (a)