符号圈拼接图的零度

令$\eta(\Gamma)$和$c(\Gamma)$是符号图$\Gamma$的零度和基本圈数. 一个符号圈拼接图是指每个块都是圈的连通符号图. 本文证明了对任意符号拼接图$\eta(\Gamma)\le c(\Gamma)+1$成立, 并且刻画了等号成立的极图, 推广了王登银等人(2022)在简单圈拼接图上的结果. 此外, 我们证明了任意的符号拼接图$\eta(\Gamma)\neq c(\Gamma)$, 给出了满足$\eta(\Gamma)=c(\Gamma)-1$的符号拼接图的一些性质并刻画处$\eta(\Gamma)=c(\Gamma)-1$的二部符号拼接图.

The Nullities of Signed Cycle-Spliced Graphs

Let $\eta(\Gamma)$ and $c(\Gamma)$ be the nullity and the cyclomatic number of a signed graph $\Gamma$. A signed cycle-spliced graph is a connected signed graph in which every block is a cycle. In this paper, we prove that for every signed cycle-spliced graph $\Gamma$, $\eta(\Gamma)\leq c(\Gamma)+1$ and the extremal graphs $\Gamma$ with nullity $c(\Gamma)+1$ are characterized, which extend the related results of Wong, Zhou and Tian (2022) on simple cycle-spliced graphs. Moreover, we prove that for every signed cycle-spliced graph $\Gamma$, $\eta(\Gamma)\neq c(\Gamma)$. Some properties on signed cycle-spliced graphs $\Gamma$ with $\eta(\Gamma)= c(\Gamma)-1$ are explored, as well as a structural characterization on signed cycle-spliced bipartite graphs $\Gamma$ satisfying $\eta(\Gamma)= c(\Gamma)-1$.