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arXiv · 2609.23707

Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph

Shaowei Sun·Mengyao Guo·Hongyan Ge·Kinkar Chandra Das
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Abstract

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.

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Shaowei Sun, Mengyao Guo, Hongyan Ge, Kinkar Chandra Das. 2026-09-20. Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph. https://arxiv.org/abs/2609.23707

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