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The Laplacian eigenvalues of mixed graphs. (English) Zbl 1017.05078

The first part of the paper gives an upper bound for the second smallest Laplacian eigenvalue of the mixed graphs, thereby generalizing the results of M. Fiedler [Czech. Math. J. 23, 298-305 (1973; Zbl 0265.05119)]. The second part presents two sharp upper bounds for the largest Laplacian eigenvalues of the mixed graphs in terms of the largest, smallest degrees and average 2-degrees, thereby improving and generalizing the main results of R. Merris [Linear Algbra Appl. 285, 33-35 (1998; Zbl 0931.05053)], and J. Li and Y. Pan [Linear Algebra Appl. 328, 153-160 (2001; Zbl 0988.05062)].

MSC:

05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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References:

[1] Bapat, R. B.; Grossman, J. W.; Kulkarni, D. M., Generalized matrix tree theorem for mixed graphs, Linear and Multilinear Algebra, 46, 299-312 (1999) · Zbl 0940.05042
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[6] Li, J. S.; Pan, Y. L., De Caen’s inequality and bounds on the largest Laplacian eigenvalue of a graph, Linear Algebra Appl., 328, 153-160 (2001) · Zbl 0988.05062
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[10] Mohar, B., Some applications of Laplace eigenvalues of graphs, (Hahn, G.; Sabidussi, G., Graph Symmetry (1997), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht), 225-275 · Zbl 0883.05096
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