Year: 2026
Type: preprint
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 $2^m+1$ 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 $(1344,221,88,26)$.
Source: arXiv (Cornell University)
Authors: Liao, Zhen-Qi, Yu, Wei-Hsuan
Institution: National Central University
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7214955330