Rank-Three Projections and Minimal Multiplicity Bipartitions of Path Complements

Abstract: For a graph (G) admitting a real symmetric realization with exactly two distinct eigenvalues, (MB(G)) is the minimum, over all such realizations, of the smaller of the two eigenvalue multiplicities. Adm, Fallat, Meagher, Nasserasr, Plosker, and Yang asked for this parameter for the complement of a path on at least eight vertices. We answer their question completely by proving $MB(\overline{P_n})=3 \qquad (n\ge 6).$ In particular, this resolves the previously unresolved orders (n\ge 9) divisible by three. The proof is exact and constructive. We exhibit six vectors in (\mathbb{R}^3) whose mutual inner products vanish exactly for consecutive indices, whose rank-one outer products form a basis of (\mathbb{S}^3), and which admit a strictly positive Parseval scaling. An elementary absorption lemma then permits any finite faithful orthogonal extension of this vector chain to be added with small positive weights while the six original weights are corrected to retain the Parseval identity. The resulting Gram matrix is a rank-three orthogonal projection in (\mathcal{S}(\overline{P_n})). A local two-dimensional orthogonality obstruction gives the matching lower bound. For completeness, we include self-contained proofs of the exceptional small orders: (MB(\overline{P_3})=1), whereas (q(\overline{P_4})=4) and (q(\overline{P_5})=3).