Infinite series of Deza graphs with strongly regular children

@misc{golubyatnikov2026infinite,
  title = {{Infinite series of Deza graphs with strongly regular children}},
  author = {Mikhail P. Golubyatnikov},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2608.13089},
  eprint = {2608.13089},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2608.13089},
  note = {CC BY 4.0},
}

TY  - UNPB
TI  - Infinite series of Deza graphs with strongly regular children
AU  - Mikhail P. Golubyatnikov
PY  - 2026
DA  - 2026/10/06
PB  - arXiv
DO  - 10.48550/arxiv.2608.13089
UR  - https://arxiv.org/abs/2608.13089
AB  - A Deza graph is a regular graph in which the number of common neighbours of two distinct vertices takes at most two values, regardless of adjacency. Its children are the graphs on the same vertex set in which adjacency is determined by these two common-neighbour counts. A Deza graph is called strongly Deza if both children are strongly regular. We construct an infinite family of edge-regular strongly Deza graphs using non-degenerate quadratic forms over the field with five elements. For every odd dimension greater than three and each of the two determinant square classes, we obtain a graph on the projective points represented by vectors of norm one. Its children are complementary strongly regular graphs with the parameters of the corresponding orthogonality graph on non-isotropic points and its complement. We determine the parameters by counting solutions to systems involving the associated bilinear form. These counts also yield symmetric association schemes over finite fields of odd characteristic. Further constructions include Deza graphs in dimension four, orthogonality graphs in odd dimensions, unions of relations in even dimensions over the field with nine elements, and an odd-dimensional family over the field with thirteen elements. We also give low-dimensional examples, including one whose children are a triangular graph and its complement.
ER  -