Graphs with three eigenvalues from group divisible designs

Abstract

We construct biregular graphs with three distinct eigenvalues from semiregular group divisible designs. For every prime power q, we use the spectral structure of Muzychuk's Construction~A on q+1 copies of AG(2,q²) to produce suitable group divisible designs. The resulting biregular graphs have 2q⁵+q⁴+q² vertices and distinct eigenvalues q⁵+q³+q², q², and −q². Their coherent closures have rank $12$ and type [3&2 2&5]. We also determine the four-class symmetric association schemes that are the coherent closures of the auxiliary graphs in this family. Computational searches provide further graphs on $81$ and $156$ vertices, using Muzychuk's Construction~B and a generalised quadrangle, respectively; their coherent closures have ranks $13$ and $10$. The $156$-vertex example has a strongly regular design as its coherent closure, of type [3&2 2&3], answering a question of the authors. We also realise a $126$-vertex example by switching a strongly regular graph.

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BibTeX RIS

BibTeX

@misc{greaves2026graphs,
  title = {{Graphs with three eigenvalues from group divisible designs}},
  author = {Gary Greaves and Jose Yip},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.21724},
  eprint = {2610.21724},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.21724},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - Graphs with three eigenvalues from group divisible designs
AU  - Gary Greaves
AU  - Jose Yip
PY  - 2026
DA  - 2026/10/15
PB  - arXiv
DO  - 10.48550/arxiv.2610.21724
UR  - https://arxiv.org/abs/2610.21724
AB  - We construct biregular graphs with three distinct eigenvalues from semiregular group divisible designs. For every prime power q, we use the spectral structure of Muzychuk's Construction~A on q+1 copies of AG(2,q^2) to produce suitable group divisible designs. The resulting biregular graphs have 2q^5+q^4+q^2 vertices and distinct eigenvalues q^5+q^3+q^2, q^2, and −q^2. Their coherent closures have rank $12$ and type [3&2 2&5]. We also determine the four-class symmetric association schemes that are the coherent closures of the auxiliary graphs in this family. Computational searches provide further graphs on $81$ and $156$ vertices, using Muzychuk's Construction~B and a generalised quadrangle, respectively; their coherent closures have ranks $13$ and $10$. The $156$-vertex example has a strongly regular design as its coherent closure, of type [3&2 2&3], answering a question of the authors. We also realise a $126$-vertex example by switching a strongly regular graph.
ER  -

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