The relations between nodal networks and edge networks are essential for the networked systems that have to consider the coupling of nodal dynamics and edge dynamics, such as social networks and power networks. In this paper, the direct relations between nodal networks and the corresponding edge networks are studied. Based on the incidence matrix and the degree matrix of a given nodal network, the Laplacian matrices of the nodal network and its edge network are formulated for undirected networks. The spectral properties of both nodal and edge Laplacian matrices are further presented. The consensus relations between the nodal dynamic systems and the edge dynamic systems are given. The convergence properties of edge networks are also presented from the spectral perspective.