The Nullities of Signed Cycle-Spliced Graphs

Abstract:

      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 $\eta(\Gamma)= c(\Gamma)-1$.