Skip to content
DraconiticResearch
Search Sources API
197,768 studies31 sourcesUpdated 16 min ago
PreprintarXiv licencearXiv

On the Brouwer-type Conjecture for Signless Laplacian Eigenvalues of Graphs

  • Ruisong Yuan
  • Xiao-Dong Zhang
Open at sourcePDFDOICite

Abstract

Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph G on n vertices with m edges, the sum Sk+(G) of its k largest signless Laplacian eigenvalues satisfies Sk+(G)≤ m+binomk+12 for k=1, …, n. In this paper, we prove that the above conjecture holds. Moreover, the equality holds if and only if k=1 and G is either star K1,a or triangle K3 with adding some isolated vertices. For split graphs, properties of block signless Laplacian matrices based on clique and independent set are adapted. While for non-split graphs, some spectral graph substructure are used to control the sum of signless Laplacian eigenvalues.

Related

  1. A solution to a conjecture on the signless Laplacian spectral radius for t-color-critical graphs
    Ming-Zhu Chen, Ya-Lei Jin, Peng-Li Zhang +1
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. The Distance Laplacian and Distance Signless Laplacian Spectra of 𝒞-Graphs
    Santanu Mandal, Pallabi Manna
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  3. The least signless Laplacian eigenvalue of {C3,C5}-free graphs
    Qi Zhou
    arXivPreprintCC0Full textCombinatorics & discrete mathematicsarXiv
  4. A sharp bound for the least signless Laplacian eigenvalue of triangle-free graphs
    Dongxiu Cai, Zhengbo Chen, Xiao-Dong Zhang
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  5. Characterization of graphs attaining the maximum signless Laplacian spectral radius under forbidden cycles and theta graphs
    Mainak Basunia, Pratima Panigrahi
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv

Cite

BibTeXRIS
BibTeX
@misc{yuan2026brouwertype,
  title = {{On the Brouwer-type Conjecture for Signless Laplacian Eigenvalues of Graphs}},
  author = {Ruisong Yuan and Xiao-Dong Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.31355},
  eprint = {2609.31355},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.31355},
}
RIS
TY  - UNPB
TI  - On the Brouwer-type Conjecture for Signless Laplacian Eigenvalues of Graphs
AU  - Ruisong Yuan
AU  - Xiao-Dong Zhang
PY  - 2026
DA  - 2026/09/25
PB  - arXiv
DO  - 10.48550/arxiv.2609.31355
UR  - https://arxiv.org/abs/2609.31355
AB  - Motivated by Brouwer's conjecture, Ashraf, Omidi and Tayfeh-Rezaie proposed the following Brouwer-type conjecture that for every graph G on n vertices with m edges, the sum S_k^+(G) of its k largest signless Laplacian eigenvalues satisfies S_k^+(G)≤ m+binomk+12 for k=1, …, n. In this paper, we prove that the above conjecture holds. Moreover, the equality holds if and only if k=1 and G is either star K_1,a or triangle K_3 with adding some isolated vertices. For split graphs, properties of block signless Laplacian matrices based on clique and independent set are adapted. While for non-split graphs, some spectral graph substructure are used to control the sum of signless Laplacian eigenvalues.
ER  -