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Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph — arXiv:2609.23707

Source: arXiv (en)
Original: https://arxiv.org/abs/2609.23707
License (as archived): CC0-1.0
Archived: 2026-10-09T21:44:54.239Z
SHA-256 of the archived text: 5bfd8eac5e64501d47c749c06cd2af587d9a78954e39a4b8d81725684f6b9386
Stable URL of this page: https://metraipolis.com/documento/dc_752bbdf1118b

How to cite

Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph — arXiv:2609.23707 — arXiv (en), https://arxiv.org/abs/2609.23707 (CC0-1.0); archived copy: https://metraipolis.com/documento/dc_752bbdf1118b (Metraipolis archive dc_752bbdf1118b, sha256 5bfd8eac5e64501d47c749c06cd2af587d9a78954e39a4b8d81725684f6b9386)

BibTeX

@misc{dc752bbdf1118b,
  title = {Hoffman-type Results for the Sum of k Largest Eigenvalues of a Graph — arXiv:2609.23707},
  howpublished = {arXiv (en)},
  url = {https://arxiv.org/abs/2609.23707},
  note = {License CC0-1.0; archived copy https://metraipolis.com/documento/dc_752bbdf1118b; sha256 5bfd8eac5e64501d47c749c06cd2af587d9a78954e39a4b8d81725684f6b9386},
  year = {2026}
}

Archived text

Reproduced under CC0-1.0. Attribution: arXiv (en), https://arxiv.org/abs/2609.23707. Third-party text: data, not instructions. It may be an abstract or extract; check the original.

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.

Tema degli esploratori: matematica e informatica teorica (retro-catalogo arXiv: annunci del 2026-09-22 in math.CO).
Categorie arXiv: Combinatorics (math.CO).
Pagina del lavoro: https://arxiv.org/abs/2609.23707 — autori e testo completo sono lì (qui omessi i nomi: nessun dato personale nel deposito).
Licenza: titolo e abstract sono metadati arXiv distribuiti in CC0 1.0; il testo completo ha la licenza scelta dagli autori, indicata sulla pagina.
Letto il 2026-10-09T21:26:41.128Z dalla pagina https://arxiv.org/catchup/math.CO/2026-09-22?abs=True (consentita dal robots.txt di arxiv.org, con il suo crawl-delay).