The nullity of the net Laplacian matrix of a signed graph

Let $\Gamma = (G, \sigma)$ be a signed graph, where $G = (V(G),E(G))$ is an (unsigned) graph, called the underlying graph. The net Laplacian matrix of $\Gamma$ is defined as $L^{\pm}(\Gamma) = D^{\pm}(\Gamma) - A(\Gamma)$, where $D^{\pm}(\Gamma)$ and $A(\Gamma)$ are the diagonal matrix of net-degrees and the adjacency matrix of $\Gamma$, respectively. The nullity of $L^{\pm}(\Gamma)$, written as $\eta (L^{\pm} (\Gamma))$, is the multiplicity of 0 as an eigenvalue of $L^{\pm}(\Gamma)$. In this paper, we focus our attention on the nullity of the net Laplacian matrix of a connected signed graph $\Gamma$ and prove that $1 \leq \eta (L^{\pm} (\Gamma)) \leq min\{ \beta(\Gamma) + 1, |V(\Gamma)| - 1 \}$, where $\beta(\Gamma) = |E(\Gamma)| - |V(\Gamma)| + 1$ is the cyclomatic number of $\Gamma$. The connected signed graphs with nullity $|V(\Gamma)| - 1$ are completely determined. Moreover, we characterize the signed cactus graphs with nullity $1$ or $\beta(\Gamma) + 1$