First eigenvalue of nonsingular mixed graphs with given number of pendant vertices

Let G be a mixed graph and L(G) be the Laplacian matrix of the mixed graph G. The first eigenvalue of G is referred to the least nonzero eigenvalue of L(G). Let MG(n,k) be the set of nonsingular mixed graphs with n vertices and k pendant vertices, where n≥4. In this paper, up to a signature matrix, we determine the unique graph with the minimal first eigenvalue among all graphs in MG(n,k). Thus we obtain a lower bound for the first eigenvalue of a mixed graph in terms of the number of pendant vertices.