The nullity of the net Laplacian matrix of a signed graph - Zhuang Xiong (The American Journal of Combinatorics)
The net Laplacian matrix of a signed graph \(Γ = (G, σ)\), where \(G = (V(G),E(G))\) is an unsigned graph (referred to as the underlying graph) and \(σ: E(G) → \{-1, +1\}\) is the sign function, is defined as \(L^{±}(Γ) = D^{±}(Γ) - A(Γ)\). Here, \(D^{±}(Γ)\) and \(A(Γ)\) represent the diagonal matrix of net-degrees and the adjacency matrix of \(Γ\), respectively. The nullity of \(L^{±}(Γ)\), denoted as \(η (L^{±} (Γ))\), refers to the multiplicity of \(0\) as an eigenvalue of \(L^{±}(Γ)\). In this paper, we concentrate on the nullity of the net Laplacian matrix of a connected signed graph \(Γ\), and establish that \(1 ≤ η (L^{±} (Γ)) ≤ \min\{ β(Γ) + 1, |V(Γ)| - 1 \}\), where \(β(Γ) = |E(Γ)| - |V(Γ)| + 1\) denotes the cyclomatic number of \(Γ\). We completely determine the connected signed graphs with nullity \(|V(Γ)| - 1\). Additionally, we characterize the signed cactus graphs with nullity \(1\) or \(β(Γ) + 1\).