AI 中文总结
研究谱图理论中具有恰好三个主特征值的图的分类问题，利用公平划分将谱条件转化为有理数上线性系统，证明直径为5的此类树的同构情况，并构造了直径无界的无限族。
AI 中文摘要
图的一个特征值若其特征空间不与全1向量正交，则称为主特征值。20世纪70年代初由Cvetković引入，Rowlinson等人对具有恰好一个或两个主特征值的图进行了系统研究，目前已充分理解。然而，具有恰好三个主特征值的图的分类仍是谱图理论中一个具有挑战性的开放问题。本文对所有直径为5且恰好具有三个主特征值的树进行了完全分类。利用公平划分，谱条件简化为有理数上线性系统的唯一可解性，导致涉及分支长度和悬垂计数的丢番图方程。我们证明，每棵这样的树要么同构于对称树\(T_r(a)\)，要么同构于由算术可除性条件确定的参数族\(\mathcal{T}\)的一个成员。我们还构造了一个直径无界的此类树的无限族。
英文摘要
An eigenvalue of a graph is called main if its eigenspace is not orthogonal to the all-ones vector. Introduced by Cvetković in the early 1970s and systematically studied by Rowlinson and others, graphs with exactly one or two main eigenvalues are now well understood. However, the classification of graphs with precisely three main eigenvalues remains a challenging open problem in spectral graph theory. This paper provides a complete classification of all trees of diameter 5 with exactly three main eigenvalues. Using equitable partitions, the spectral condition reduces to the unique solvability of linear systems over the rationals, leading to Diophantine equations involving branch lengths and pendant counts. We prove that every such tree is isomorphic either to a symmetric tree $T_r(a)$ or to a member of a parametric family $\mathcal{T}$ determined by arithmetic divisibility conditions. We also construct an infinite family of such trees with unbounded diameter.
Comments18 pages
