Geodesic
On eigenvectors of mixed graphs with exactly one nonsingular cycle
Czechoslovak Mathematical Journal, Tome 57 (2007) no. 4, pp. 1215-1222 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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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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/

[1] R. B.  Bapat, S.  Pati: Algebraic connectivity and the characteristic set of a graph. Linear Multilinear Algebra 45 (1998), 247–273. | DOI | MR

[2] R. B.  Bapat, J. W.  Grossman, D. M.  Kulkarni: Generalized matrix tree theorem for mixed graphs. Linear Multilinear Algebra 46 (1999), 299–312. | DOI | MR

[3] R. B.  Bapat, J. W.  Grossman, D. M.  Kulkarni: Edge version of the matrix tree theorem for trees. Linear Multilinear Algebra 47 (2000), 217–229. | DOI | MR

[4] J. A.  Bondy, U. S. R.  Murty: Graph Theory with Applications. Elsevier, New York, 1976. | MR

[5] F. R. K.  Chung: Spectral Graph Theory. Conference Board of the Mathematical Science, No.  92. Am. Math. Soc., Providence, 1997. | MR

[6] Y.-Z.  Fan: On spectra integral variations of mixed graph. Linear Algebra Appl. 374 (2003), 307–316. | DOI | MR

[7] M.  Fiedler: Algebraic connectivity of graphs. Czechoslovak Math.  J. 23 (1973), 298–305. | MR | Zbl

[8] M.  Fiedler: A property of eigenvectors of nonnegative symmetric matrices and its applications to graph theory. Czechoslovak Math.  J. 25 (1975), 619–633. | MR

[9] J. W.  Grossman, D. M.  Kulkarni, I. E.  Schochetman: Algebraic graph theory without orientations. Linear Algebra Appl. 212/213 (1994), 289–308. | MR

[10] J. M.  Guo, S. W.  Tan: A relation between the matching number and the Laplacian spectrum of a graph. Linear Algebra Appl. 325 (2001), 71–74. | MR

[11] R. A.  Horn, C. R.  Johnson: Matrix Analysis. Cambridge Univ. Press, Cambridge, 1985. | MR

[12] S.  Kirkland, S.  Fallat: Perron components and algebraic connectivity for weighted graphs. Linear Multilinear Algebra 44 (1998), 131–148. | DOI | MR

[13] S.  Kirkland, M.  Neumann, B.  Shader: Characteristic vertices of weighted trees via Perron values. Linear Multilinear Algebra 40 (1996), 311–325. | DOI | MR

[14] L. S.  Melnikov, V. G.  Vizing: The edge chromatic number of a directed/mixed multi-graph. J.  Graph Theory 31 (1999), 267–273. | DOI | MR

[15] R.  Merris: Laplacian matrices of graphs: a survey. Linear Algebra Appl. 197/198 (1994), 143–176. | MR | Zbl

[16] B.  Mohar: Some applications of Laplacian eigenvalues of graphs. In: Graph Symmetry, G. Hahn and G. Sabidussi (eds.), Kluwer, Dordrecht, 1997, pp. 225–275. | MR

[17] X.-D.  Zhang, J.-S.  Li: The Laplacian spectrum of a mixed graph. Linear Algebra Appl. 353 (2002), 11–20. | MR | Zbl