AI 中文总结
本文研究奇围约束下图的最大与最小特征值之和的渐近上界，通过抽象矩问题证明其上确界介于$(1-o(1))(\log g)^2/g^3$和$(10/7+o(1))(\log g)^2/g^3$之间，并对正则图给出最优界。
AI 中文摘要
奇围以及最大与最小邻接特征值之和是衡量图与二部图接近程度的两个常用指标。本文研究了在奇围约束下该特征值之和的渐近行为。我们研究了相关的抽象矩问题，该问题涉及满足奇围至少为$g$所强加的消失奇矩恒等式以及图谱的标准二阶矩界的有限实数列表。我们证明，当$g$通过奇数趋于无穷时，这些列表上$(\u03bb_1+\u03bb_n)/n$的上确界介于$(1-o(1))(\log g)^2/g^3$和$(10/7+o(1))(\log g)^2/g^3$之间。上界特别适用于奇围至少为$g$的图。对于奇围$g\geq5$的正则图，我们得到更强的界$(\u03bb_1+\u03bb_n)/n \leq 12/((g-1)(g-2)(g-3))$，该界在常数因子意义下是最优的。
英文摘要
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 $(λ_1+λ_n)/n$ over these lists lies between $(1-o(1))(\log g)^2/g^3$ and $(10/7+o(1))(\log g)^2/g^3$ as $g\to\infty$ through the odd integers. The upper bound applies in particular to graphs of odd girth at least $g$. For regular graphs of odd girth $g\geq5$, we obtain a stronger bound $(λ_1+λ_n)/n \leq 12/((g-1)(g-2)(g-3))$, which is optimal up to a constant factor.
