In this paper, a new approach for finding the Hamiltonian circuit of a graph is presented. This approach relies for the first time in literature on a matrix constructed from the mesh-links incidence matrix of the graph, in a similar way as the Laplacian is constructed from the nodes-links incidence matrix. The method is very efficient and has the advantage of easier verification steps than comparable related work. Some examples for which the complexity of the method is polynomial, are shown. The approach is useful in finding the ground loops and ground traces of a circuit.