论文标题
无签名的图形$η（g，σ）= | v（g）| -2m（g）+2c（g）-1 $
No signed graph with the nullity $η(G,σ)=|V(G)|-2m(G)+2c(G)-1$
论文作者
论文摘要
令$ g^σ=（g，σ）$为签名图，$ a（g，σ）$为其邻接矩阵。用$ m（g）$表示$ g $的匹配数。令$η（g，σ）$为$（g，σ）$的无效。他等人。 [根据等级的匹配数和循环数的边界，线性代数应用。 572（2019），273--291]证明了$$ | v（g）| -2m（g）-c（g）-c（g）\leqη（g，σ）\ leq | v（g）| -2m（g）+2c（g）+2c（g），$ c（g），$ c（g）$ c（g）$是$ g $的周期空间。到达下限或上限的签名图分别以同一论文为特征。在本文中，我们将证明没有签名的图形$ | v（g）| -2m（g）+2c（g）-1 $。我们还证明，对于给定的$ c（g）$，有无限签名的图形，$ | v（g）| -2m（g）+2c（g）+2c（g）-s，〜（0 \ leq s \ leq 3c（g），s \ neq1）$。
Let $G^σ=(G,σ)$ be a signed graph and $A(G,σ)$ be its adjacency matrix. Denote by $m(G)$ the matching number of $G$. Let $η(G,σ)$ be the nullity of $(G,σ)$. He et al. [Bounds for the matching number and cyclomatic number of a signed graph in terms of rank, Linear Algebra Appl. 572 (2019), 273--291] proved that $$|V(G)|-2m(G)-c(G)\leqη(G,σ)\leq |V(G)|-2m(G)+2c(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 no signed graphs with nullity $|V(G)|-2m(G)+2c(G)-1$. We also prove that there are infinite signed graphs with nullity $|V(G)|-2m(G)+2c(G)-s,~(0\leq s\leq3c(G), s\neq1)$ for a given $c(G)$.
