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PreprintCC BY-NC-ND 4.0arXiv

Unbalanced distance-biregular graphs with girth deficiency two

  • Sebastiano Argenti
  • Alessandro Siciliano
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Abstract

Let Γ=(V,E) be a distance-biregular graph with even diameter d and girth 2d−2. On the edge-set E we define relations that form an association scheme; this is done by considering the linear span of the corresponding adjacency matrices, and, by means of intersection diagrams, we prove that this is an algebra. Furthermore, we describe all the irreducible representations of this algebra, and, by considering the multiplicities of the 2-dimensional ones, we show that there are no such distance-biregular graphs with 28≤ d≤ 46 and d≥ 52.

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Cite

BibTeXRIS
BibTeX
@misc{argenti2026unbalanced,
  title = {{Unbalanced distance-biregular graphs with girth deficiency two}},
  author = {Sebastiano Argenti and Alessandro Siciliano},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.07817},
  eprint = {2609.07817},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.07817},
  note = {CC BY-NC-ND 4.0},
}
RIS
TY  - UNPB
TI  - Unbalanced distance-biregular graphs with girth deficiency two
AU  - Sebastiano Argenti
AU  - Alessandro Siciliano
PY  - 2026
DA  - 2026/09/07
PB  - arXiv
DO  - 10.48550/arxiv.2609.07817
UR  - https://arxiv.org/abs/2609.07817
AB  - Let Γ=(V,E) be a distance-biregular graph with even diameter d and girth 2d−2. On the edge-set E we define relations that form an association scheme; this is done by considering the linear span of the corresponding adjacency matrices, and, by means of intersection diagrams, we prove that this is an algebra. Furthermore, we describe all the irreducible representations of this algebra, and, by considering the multiplicities of the 2-dimensional ones, we show that there are no such distance-biregular graphs with 28≤ d≤ 46 and d≥ 52.
ER  -