In this paper, we examine the properties of the Laplacian matrix defined on signed networks, referred to as the signed Laplacian matrix, from a graph-theoretic perspective. The connection between the stability of the signed Laplacian with the cut set of the network is established. This is then followed by relating and the number of negative eigenvalues of the signed Laplacian to the number of negatively weighted edges in the network. In order to stabilize the signed Laplacian dynamics, a distributed diagonal compensation approach is proposed; we show that this compensation is closely related to the structural balance of the network. Furthermore, the influence of the external input exerted on the signed Laplacian dynamics is investigated.