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PreprintarXiv licencearXiv

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 Sk(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 Sk(G). For each fixed k≥ 2 and sufficiently large order n, we characterize all connected graphs satisfying Sk(G)<2k. As a consequence, we prove that the path Pn is the unique minimizer of Sk(G) among all connected graphs of order n. \vspace*{2mm} We further investigate the first Hoffman-type range 2k≤ Sk(G)<2k+√(2+√5)−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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Cite

BibTeXRIS
BibTeX
@misc{sun2026hoffmantype,
  title = {{Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph}},
  author = {Shaowei Sun and Mengyao Guo and Hongyan Ge and Kinkar Chandra Das},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23707},
  eprint = {2609.23707},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23707},
}
RIS
TY  - UNPB
TI  - Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph
AU  - Shaowei Sun
AU  - Mengyao Guo
AU  - Hongyan Ge
AU  - Kinkar Chandra Das
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23707
UR  - https://arxiv.org/abs/2609.23707
AB  - 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≥ 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≤ S_k(G)<2k+√(2+√5)−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.
ER  -