For a graph G, let S(G) be the set of real symmetric matrices whose off-diagonal zero–nonzero pattern is the adjacency pattern of G. Define MB(G) to be the minimum, over all A∈S(G) having exactly two distinct eigenvalues, of the smaller of their two multiplicities. Determine MB(overlineP_n) for every integer n≥ 8, where P_n is the path on n vertices.

Adm–Fallat–Meagher–Nasserasr–Plosker–Yang question on MB $\overline{P_n}$

For a graph $G$ , let $\mathcal{S}(G)$ be the set of real symmetric matrices whose off-diagonal zero–nonzero pattern is the adjacency pattern of $G$ . Define $MB(G)$ to be the minimum, over all $A\in\mathcal{S}(G)$ having exactly two distinct eigenvalues, of the smaller of their two multiplicities. Determine $MB(\overline{P_n})$ for every integer $n\ge 8$ , where $P_n$ is the path on $n$ vertices.

References

Primary source

arXiv

Additional references

Rank-Three Projections and Minimal Multiplicity Bipartitions of Path Complements — arXiv

Posted 1 month ago

Progress summary

Refreshed 1 month ago

Claimed solved

An August 2026 preprint claims to determine the remaining path-complement cases, but the result has not been independently checked.

The question asks for the multiplicity bipartition of the complement of a path on at least $8$ vertices. The 2019 source records $MB(P_6)=MB(P_7)=3$ as the only known values then and explicitly poses the unresolved general question.

August 27, 2026 preprint

A report dated August 27, 2026, says that Rank-Three Projections and Minimal Multiplicity Bipartitions of Path Complements claims the exact value for all $n \ge 6$ and records the exceptional small cases, thereby claiming to close the infinite family. The claim is unverified.

Current status (as of August 2026): The question is claimed solved for $n \ge 6$ , with exceptional small cases stated, but the purported resolution remains independently unverified.

Sources

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