On the Definiteness and the Second Smallest Eigenvalue of Signed Laplacian Matrices

In-depth understanding of the definiteness of signed Laplacian matrices is critical for the analysis of the cooperative behavior of dynamical systems. In this letter, we focus on undirected signed weighted graphs and prove that the signed Laplacian matrix has at most $p$ negative eigenvalues for a graph with $p$ negative edges, and for the signed Laplacian corresponding to a tree, it has exactly $p$ negative eigenvalues. An algebraic necessary and sufficient condition for the positive semi-definiteness of the signed Laplacians with a simple zero eigenvalue is given. Furthermore, we establish the upper and lower bounds for the second smallest eigenvalue of the signed Laplacians when adding a negative edge to a pair of non-adjacent vertices and show that these bounds are tight.