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First eigenvalue and first eigenvectors of a nonsingular unicyclic mixed graph |
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| Authors: | Yi-Zheng Fan Shi-Cai Gong Yi Wang Yu-Bin Gao |
| Institution: | a Key Laboratory of Intelligent Computing & Signal Processing, Ministry of Education of the People’s Republic of China, Anhui University, Hefei 230039, PR China b School of Mathematics and Computation Sciences, Anhui University, Hefei 230039, PR China c Department of Mathematics, North University of China, Taiyuan 030051, PR China |
| Abstract: | Let G be a mixed graph and let L(G) be the Laplacian matrix of the graph G. The first eigenvalue and the first eigenvectors of G are respectively referred to the least nonzero eigenvalue and the corresponding eigenvectors of L(G). In this paper we focus on the properties of the first eigenvalue and the first eigenvectors of a nonsingular unicyclic mixed graph (abbreviated to a NUM graph). We introduce the notion of characteristic set associated with the first eigenvectors, and then obtain some results on the sign structure of the first eigenvectors. By these results we determine the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order and fixed girth, and the unique graph which minimizes the first eigenvalue over all NUM graphs of fixed order. |
| Keywords: | Mixed graph Unicyclic graph Eigenvector Eigenvalue Characteristic set Girth |
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