强正则图不存在性的多点球面半定界

中文摘要 AI 辅助

我们使用多点半定规划来证明强正则图的不存在性。一个本原强正则图的归一化特征空间投影给出了一个球面两点距离集，每个顶点对应一个点。因此，一个小于所需点数的界即可排除该图。遵循de Laat等人以及Kao和Yu的公式，我们针对每种参考类型，使用由所选点的可行内积标签索引的系数矩阵。在零次项中，我们将由参考点索引的矩阵坐标合并为单一坐标。所得程序使用最多六个点的配置，对于包含m个点的参考集，矩阵阶数至多为$2^m+1$。通过精确有理算术验证的可行对偶解，排除了参数为$(351,140,73,44)$、$(550,162,75,36)$、$(703,182,81,35)$和$(1344,221,88,26)$的强正则图。

英文摘要

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 $(1344,221,88,26)$.

发表机构

• National Central University（中央大学）

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