直径三的广义奇图中的谱二部性
Spectral bipartiteness in generalized odd graphs of diameter three
- School of Finance and Mathematics, Huainan Normal University（淮南师范学院金融与数学学院）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究确定了直径为三、奇围长至少七的非二部距离正则图中谱不变量σ(G)的前三个最大值，分别由折叠7立方体、奇图O_4和C_7取得，并证明其余图均小于1/36，解决了相关公开问题。
中文摘要 AI 辅助
对于阶为$n$的图$G$，令$\sigma(G)=(\lambda_1(G)+\lambda_n(G))/n$。我们确定了直径为三、奇围长至少为七的非二部距离正则图中该不变量的前三个最大值。唯一达到最大值的图是折叠$7$立方体，其值为$1/32$；唯一达到第二大的图是奇图$O_4$，其值为$1/35$；唯一达到第三大的图是$C_7$，其值为$2(1-\cos(\pi/7))/7$。更精确地说，该类中所有其他图都满足$\sigma(G)<1/36$。这回答了Abiad、Taranchuk和van Veluw在《Electronic Journal of Combinatorics》33(2) (2026), P2.31中提出的问题11。证明结合了已有的局部重数和奇矩界：条件$\sigma(G)\geq1/36$迫使价数至多为$182$。随后仅使用整数和有理数算术的穷举证书留下三个交集数组。完整证书公开可用，且既不假设广义奇图的分类，也不假设$Q$-多项式性质。奇围长定理对具有至多四个不同邻接特征值的连通无$\{C_3,C_5\}$图给出了相同的极值结论，而无需假设正则性。
英文摘要
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.
