The nullity of the net Laplacian matrix of a signed graph

Abstract

The net Laplacian matrix of a signed graph \(\Gamma = (G, \sigma)\), where \(G = (V(G),E(G))\) is an unsigned graph (referred to as the underlying graph) and \(\sigma: E(G) \rightarrow \{-1, +1\}\) is the sign function, is defined as \(L^{\pm}(\Gamma) = D^{\pm}(\Gamma) - A(\Gamma)\). Here, \(D^{\pm}(\Gamma)\) and \(A(\Gamma)\) represent the diagonal matrix of net-degrees and the adjacency matrix of \(\Gamma\), respectively. The nullity of \(L^{\pm}(\Gamma)\), denoted as \(\eta (L^{\pm} (\Gamma))\), refers to the multiplicity of \(0\) as an eigenvalue of \(L^{\pm}(\Gamma)\). In this paper, we concentrate on the nullity of the net Laplacian matrix of a connected signed graph \(\Gamma\), and establish that \(1 \leq \eta (L^{\pm} (\Gamma)) \leq \min\{ \beta(\Gamma) + 1, |V(\Gamma)| - 1 \}\), where \(\beta(\Gamma) = |E(\Gamma)| - |V(\Gamma)| + 1\) denotes the cyclomatic number of \(\Gamma\). We completely determine the connected signed graphs with nullity \(|V(\Gamma)| - 1\). Additionally, we characterize the signed cactus graphs with nullity \(1\) or \(\beta(\Gamma) + 1\).