Skip to content
DraconiticResearch
Search Sources API
197,357 studies31 sourcesUpdated 38 min ago
PreprintCC BY 4.0arXiv

Neumaier graphs of coherent rank five

  • Gary Greaves
  • Zhao Kuang Tan
Open at sourcePDFDOICite

Abstract

We construct an infinite family of Neumaier graphs of coherent rank five, answering the existence question at the smallest possible coherent rank beyond the strongly regular case. For every prime power qgeqslant7 with q≡3pmod4, set n=q+1. Each graph in our construction has precisely five distinct eigenvalues, Neumaier parameters ( n(n−1)(n−3), (n2(n−3))/2, (n(n2−n−8))/4; ((n−2)2)/2, (n−1)(n−3) ), and its adjacency matrix lies in the Bose--Mesner algebra of the four-class association scheme of Holzmann, Kharaghani, and Suda. Paley Hadamard matrices and Desarguesian mutually orthogonal Latin squares yield an infinite family whose smallest member has parameters $(280,160,96;18,35)$.

Related

  1. A unified framework for existing and new constructions of Neumaier graphs
    Aida Abiad, Wouter Castryck, Maarten De Boeck +2
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  2. On the hit problem of rank five
    Nguyen Sum
    arXivPreprintCC BY 4.0Full textGeometry & topologyarXiv
  3. Perfect matchings in planar 3-connected graphs: girth five and minimum degree five
    Xinyu Dai
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsarXiv
  4. Sharp Euler-factor positivity for the BG-rank-refined five-core crank
    Huan Xiong
    arXivPreprintCC BY 4.0Full textCombinatorics & discrete mathematicsarXiv
  5. Stabilizer entropy cones as self-dual slices of linear rank cones: a five-party gap and dependence on the local dimension
    Xidan Song, Ruifeng Cao
    arXivPreprintarXiv licenceFull textCombinatorics & discrete mathematicsQuantum physics & quantum informationarXiv

Cite

BibTeXRIS
BibTeX
@misc{greaves2026neumaier,
  title = {{Neumaier graphs of coherent rank five}},
  author = {Gary Greaves and Zhao Kuang Tan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.02218},
  eprint = {2609.02218},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.02218},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Neumaier graphs of coherent rank five
AU  - Gary Greaves
AU  - Zhao Kuang Tan
PY  - 2026
DA  - 2026/09/02
PB  - arXiv
DO  - 10.48550/arxiv.2609.02218
UR  - https://arxiv.org/abs/2609.02218
AB  - We construct an infinite family of Neumaier graphs of coherent rank five, answering the existence question at the smallest possible coherent rank beyond the strongly regular case. For every prime power qgeqslant7 with q≡3pmod4, set n=q+1. Each graph in our construction has precisely five distinct eigenvalues, Neumaier parameters ( n(n−1)(n−3), (n^2(n−3))/2, (n(n^2−n−8))/4; ((n−2)^2)/2, (n−1)(n−3) ), and its adjacency matrix lies in the Bose--Mesner algebra of the four-class association scheme of Holzmann, Kharaghani, and Suda. Paley Hadamard matrices and Desarguesian mutually orthogonal Latin squares yield an infinite family whose smallest member has parameters $(280,160,96;18,35)$.
ER  -