发表机构
School of Science, Zhejiang University of Science and Technology; Department of Mathematics, Sungkyunkwan University（浙江科技学院理学院; 成均馆大学数学系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对图的k个最大特征值之和，提出了加性Hoffman型问题，刻画了满足S_k(G)<2k的连通图，证明路径图是唯一极小化者，并进一步刻画了第一个Hoffman型区间内的非树图及树的显式族。
AI 中文摘要
设$S_k(G)$表示图$G$的$k$个最大特征值之和。受经典的关于图谱半径的Hoffman程序的启发，我们研究了$S_k(G)$的加性Hoffman型问题。对于每个固定的$k\geq 2$和足够大的阶数$n$，我们刻画了所有满足$S_k(G)<2k$的连通图。作为推论，我们证明了在阶数为$n$的所有连通图中，路径$P_n$是$S_k(G)$的唯一极小化图。我们进一步研究了第一个Hoffman型区间\\[ 2k\leq S_k(G)<2k+\sqrt{2+\sqrt5}-2. \\] 我们完全刻画了该区间内的非树图，并将树的情况归结为几个显式族。证明结合了Ky Fan变分原理、来自顶点不相交子图的谱估计、小谱半径图的结构结果以及有界度图的长路径论证。
英文摘要
Let $S_k(G)$ denote the sum of the $k$ largest eigenvalues of a graph $G$. Motivated by the classical Hoffman program for the spectral radius of a graph, we investigate an additive Hoffman-type problem for $S_k(G)$. For each fixed $k\geq 2$ and sufficiently large order $n$, we characterize all connected graphs satisfying $S_k(G)<2k$. As a consequence, we prove that the path $P_n$ is the unique minimizer of $S_k(G)$ among all connected graphs of order $n$. \vspace*{2mm} We further investigate the first Hoffman-type range \[ 2k\leq S_k(G)<2k+\sqrt{2+\sqrt5}-2. \] We completely characterize the non-tree graphs in this range and reduce the tree case to several explicit families. The proofs combine Ky Fan's variational principle, spectral estimates from vertex-disjoint subgraphs, structural results for graphs with small spectral radius, and long-path arguments for bounded-degree graphs.
Comments26 pages
