Geodesic
On eigenvectors of mixed graphs with exactly one nonsingular cycle
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1215-1222.

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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.
Classification : 05C50, 15A18
Keywords: mixed graphs; Laplacian eigenvectors 
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     title = {On eigenvectors of mixed graphs with exactly one nonsingular cycle},
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Fan, Yi-Zheng. On eigenvectors of mixed graphs with exactly one nonsingular cycle. Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1215-1222. http://geodesic.mathdoc.fr/item/CMJ_2007__57_4_a7/