Yi-Zheng Fan, School of Mathematics and Computation Sciences, Key Laboratory of Intelligent Computing and Signal Processing, Ministry of Education of the People's Republic of China, Anhui University, Hefei, Anhui 230039, P. R. China, e-mail: fanyz@ahu.edu.cn
Abstract: 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.
Keywords: mixed graphs, Laplacian eigenvectors
Classification (MSC 2000): 05C50, 15A18
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