On eigenvectors of mixed graphs with exactly one nonsingular cycle

summary: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