Spectral bipartiteness in generalized odd graphs of diameter three

Abstract: For a graph $G$ of order $n$, put $σ(G)=(λ_1(G)+λ_n(G))/n$. We determine the first three largest values of this invariant among nonbipartite distance-regular graphs of diameter three and odd girth at least seven. The unique maximizer is the folded $7$-cube, with value $1/32$; the unique second maximizer is the Odd graph $O_4$, with value $1/35$; and the unique third maximizer is $C_7$, with value $2(1-\cos(π/7))/7$. More precisely, every other graph in the class satisfies $σ(G)<1/36$. This answers Problem~11 of Abiad, Taranchuk and van Veluw in \emph{Electronic Journal of Combinatorics} 33(2) (2026), P2.31. The proof combines established local multiplicity and odd-moment bounds: the condition $σ(G)\geq1/36$ forces the valency to be at most $182$. An exhaustive certificate using only integer and rational arithmetic then leaves three intersection arrays. The complete certificate is publicly available, and neither a classification of generalized odd graphs nor the $Q$-polynomial property is assumed. The odd-girth theorem gives the same extremal conclusions for connected ${C_3,C_5}$-free graphs with at most four distinct adjacency eigenvalues, without assuming regularity.