Directed strongly regular graphs with rank 5
Directed strongly regular graphs with rank 5
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DOI：
10.1016/j.laa.2015.03.019
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发表时间：
2015-07
影响因子：
1.1
通讯作者：
L. K. Jørgensen
中科院分区：
文献类型：
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作者：
L. K. Jørgensen
从有向强正则图的参数(n，k，t，λ，μ)出发，杜瓦尔(1988年)[4]证明了如何计算邻接矩阵的特征值和重数，从而计算邻接矩阵的秩.对于每个有理数q，其中1 5≤q≤7 10，存在一个秩为5且kn=q的DSRG的可行(即满足Duval条件)参数集.本文证明了只有当kn为1 5，1 3，2 5，1 2，3 5，2 3，3 5或2 3时，才存在DSRG，因此每个秩为5的DSRG都有已知图的参数.该证明基于条目在{0，1}中的5×5矩阵的枚举。
From the parameters (n, k, t, λ, μ) of a directed strongly regular graph (dsrg) A. Duval (1988)[4] showed how to compute the eigenvalues and multiplicities of the adjacency matrix, and thus the rank of the adjacency matrix. For every rational number q, where 1 5≤ q≤ 7 10, there is a feasible (ie, satisfying Duval's conditions) parameter set for a dsrg with rank 5 and with k n= q. In this paper we show that there exist a dsrg with such a feasible parameter set only if k n is 1 5, 1 3, 2 5, 1 2, 3 5, or 2 3. Every dsrg with rank 5 therefore has parameters of a known graph. The proof is based on an enumeration of 5× 5 matrices with entries in {0, 1}.