Nonexistence of strongly regular graphs via multipoint spherical semidefinite bounds
Abstract
We use multipoint semidefinite programming to prove the nonexistence of strongly regular graphs. A normalized eigenspace projection of a primitive strongly regular graph gives a spherical two-distance set with one point for each vertex. A bound smaller than the required number of points therefore rules out the graph. Following the formulations of de Laat et al. and Kao and Yu, we use, for each reference type, coefficient matrices indexed by the feasible inner-product labels of the selected points. In degree zero, we combine the matrix coordinates indexed by the reference points into a single coordinate. The resulting programs use configurations of up to six points, with matrix orders at most 2m+1 for a reference set of m points. Feasible dual solutions, verified in exact rational arithmetic, exclude strongly regular graphs with parameters $(351,140,73,44)$, $(550,162,75,36)$, $(703,182,81,35)$ and $(1344,221,88,26)$.
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BibTeX
@misc{liao2026nonexistence,
title = {{Nonexistence of strongly regular graphs via multipoint spherical semidefinite bounds}},
author = {Zhen-Qi Liao and Wei-Hsuan Yu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.37901},
eprint = {2609.37901},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.37901},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Nonexistence of strongly regular graphs via multipoint spherical semidefinite bounds
AU - Zhen-Qi Liao
AU - Wei-Hsuan Yu
PY - 2026
DA - 2026/09/29
PB - arXiv
DO - 10.48550/arxiv.2609.37901
UR - https://arxiv.org/abs/2609.37901
AB - We use multipoint semidefinite programming to prove the nonexistence of strongly regular graphs. A normalized eigenspace projection of a primitive strongly regular graph gives a spherical two-distance set with one point for each vertex. A bound smaller than the required number of points therefore rules out the graph. Following the formulations of de Laat et al. and Kao and Yu, we use, for each reference type, coefficient matrices indexed by the feasible inner-product labels of the selected points. In degree zero, we combine the matrix coordinates indexed by the reference points into a single coordinate. The resulting programs use configurations of up to six points, with matrix orders at most 2^m+1 for a reference set of m points. Feasible dual solutions, verified in exact rational arithmetic, exclude strongly regular graphs with parameters $(351,140,73,44)$, $(550,162,75,36)$, $(703,182,81,35)$ and $(1344,221,88,26)$.
ER -