Switched symplectic graphs and their 2-ranks
Switched symplectic graphs and their 2-ranks
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交换辛图及其 2 阶
DOI：
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发表时间：
2014
期刊：
影响因子：
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通讯作者：
W. Haemers
中科院分区：
文献类型：
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作者：
A. Abiad;W. Haemers
我们将Godsil-McKay切换应用于$$mathbb {F}_2$$ F2上至少63个顶点的辛图，并证明切换后图的2秩（邻接矩阵）增加。这表明，当$$
u ge 3$$ ν≥3时，切换图是一个新的强正则图，其参数为$$(2^{2
u }-1, 2^{2
u -1}, 2^{2
u -2},2^{2
u -2})$$ (22ν-1,22ν-1,22ν-2,22ν-2)，2阶为$$2
u +2$$ 2ν+2。对于63个顶点的辛图，我们用计算机研究了重复交换，发现了许多新的具有上述参数的强正则图，对于$$
u =3$$ ν=3具有不同的2阶。利用这些结果和Hadamard矩阵辛图的递归构造方法，我们得到了几个具有上述参数的图，但对于每个$$
u ge 3$$ ν≥3都有不同的2-rank。
We apply Godsil–McKay switching to the symplectic graphs over $$mathbb {F}_2$$F2 with at least 63 vertices and prove that the 2-rank of (the adjacency matrix of) the graph increases after switching. This shows that the switched graph is a new strongly regular graph with parameters $$(2^{2
u }-1, 2^{2
u -1}, 2^{2
u -2},2^{2
u -2})$$(22ν-1,22ν-1,22ν-2,22ν-2) and 2-rank $$2
u +2$$2ν+2 when $$
u ge 3$$ν≥3. For the symplectic graph on 63 vertices we investigate repeated switching by computer and find many new strongly regular graphs with the above parameters for $$
u =3$$ν=3 with various 2-ranks. Using these results and a recursive construction method for the symplectic graph from Hadamard matrices, we obtain several graphs with the above parameters, but different 2-ranks for every $$
u ge 3$$ν≥3.