On eigenvectors of mixed graphs with exactly one nonsingular cycle

Let G be a mixed graph. The eigenvalues and eigenvectors of G are respectively defined to be those of its Laplacian matrix. If G is a simple graph, [M. Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory, Czechoslovak Math. J. 25 (1975), 619–633] gave a remarkable result on the structure of the eigenvectors of G corresponding to its second smallest eigenvalue (also called the algebraic connectivity of G). For G being a general mixed graph with exactly one nonsingular cycle, using Fiedler’s result, we obtain a similar result on the structure of the eigenvectors of G corresponding to its smallest eigenvalue. Supported by National Natural Science Foundation of China (10601001), Anhui Provincial Natural Science Foundation (050460102), NSF of Department of Education of Anhui province (2004kj027), the project of innovation team on basic mathematics of Anhui University, and the project of talents group construction of Anhui University.