Asymptotic Spectral Bounds under Odd-Girth Constraints

Odd girth and the sum of the largest and smallest adjacency eigenvalues are two common indicators of how close a graph is to being bipartite. This paper investigates the asymptotic behavior of this eigenvalue sum under odd-girth constraints. We study the associated abstract moment problem over finite real lists satisfying the vanishing odd-moment identities imposed by odd girth at least $g$ and the standard second-moment bound for graph spectra. We prove that the supremum of $({\lambda }_1+{\lambda }_n)\mathrm{/}n$ over these lists lies between $(1-o(1))(\log g{)}^2\mathrm{/}{g}^3$ and $(10\mathrm{/}7+o(1))(\log g{)}^2\mathrm{/}{g}^3$ as $g\to \mathrm{\infty }$ through the odd integers. The upper bound applies in particular to graphs of odd girth at least $g$. For regular graphs of odd girth $g\ge 5$, we obtain a stronger bound $({\lambda }_1+{\lambda }_n)\mathrm{/}n\le 12\mathrm{/}((g-1)(g-2)(g-3))$, which is optimal up to a constant factor.