图切换、2秩和图形Hadamard矩阵
Graph switching, 2-ranks, and graphical Hadamard matrices
摘要：
本文研究了图的邻接矩阵在Seidel和Godsil-McKay切换下的2-秩的性质，并将所得结果应用于由$4^m$阶的图Hadamard矩阵生成的图.从已知的Hadamard矩阵的顺序$64$的图形开始，我们发现（通过计算机）许多Godsil-McKay开关集，增加2-秩。因此，我们发现强正则图的参数$(63,32,16,16)$，$(64,36,20,20)$，和$(64,28,12,12)$几乎所有可行的2-秩。此外，我们还研究了Hadamard矩阵的Kronecker积的图积的2-秩的性质，这使我们能够找到许多$4^m$阶的图Hadamard矩阵，对于这些矩阵，相关的强正则图具有无限多个不同的2-秩。本文推广了第一作者和最后一作者的文章“交换辛图及其2-秩”的结果。
英文摘要：
We study the behaviour of the 2-rank of the adjacency matrix of a graph under Seidel and Godsil-McKay switching, and apply the result to graphs coming from graphical Hadamard matrices of order $4^m$. Starting with graphs from known Hadamard matrices of order $64$, we find (by computer) many Godsil-McKay switching sets that increase the 2-rank. Thus we find strongly regular graphs with parameters $(63,32,16,16)$, $(64,36,20,20)$, and $(64,28,12,12)$ for almost all feasible 2-ranks. In addition we work out the behaviour of the 2-rank for a graph product related to the Kronecker product for Hadamard matrices, which enables us to find many graphical Hadamard matrices of order $4^m$ for which the related strongly regular graphs have an unbounded number of different 2-ranks. The paper extends results from the article 'Switched symplectic graphs and their 2-ranks' by the first and the last author.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
