No signed graph with the nullity η(G,σ)=|V(G)| − 2m(G)+2c(G)−1
No signed graph with the nullity η(G,σ)=|V(G)| − 2m(G)+2c(G)−1
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DOI：
10.1016/j.laa.2021.01.002
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发表时间：
2021-04
影响因子：
1.1
通讯作者：
Yong Lu;Jingwen Wu
中科院分区：
文献类型：
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作者：
Yong Lu;Jingwen Wu
设Gσ=(G，σ)是一个带符号图，A(G，σ)是它的邻接矩阵。用m(G)表示G的匹配个数，设η(G，σ)是(G，σ)的零性。他等人(2019)[6]证明了|V(G)|−2 m(G)−c(G)≤η(G，σ)≤|V(G)|−2 m(G)+2 c(G)，其中c(G)是G的圈空间的维度。证明了不存在零度为|V(G)|−2 m(G)+2 c(G)−1的有符号图，并证明了对于给定的c(G)，存在无穷多个零度为|V(G)|−2 m(G)+2 c(G)−S，(0≤S≤3 c(G)，S≠1)的有符号图.
Let G σ=(G, σ) be a signed graph and A (G, σ) be its adjacency matrix. Denote by m (G) the matching number of G. Let η (G, σ) be the nullity of (G, σ). He et al.(2019)[6] proved that| V (G)|− 2 m (G)− c (G)≤ η (G, σ)≤| V (G)|− 2 m (G)+ 2 c (G), where c (G) is the dimension of cycle space of G. Signed graphs reaching the lower bound or the upper bound are respectively characterized by the same paper. In this paper, we will prove that there are no signed graphs with nullity| V (G)|− 2 m (G)+ 2 c (G)− 1. We also prove that there are infinitely many signed graphs with nullity| V (G)|− 2 m (G)+ 2 c (G)− s,(0≤ s≤ 3 c (G), s≠ 1) for a given c (G).