A note on Merris’ conjectures. (English) Zbl 0940.05045
Some conjectures of R. Merris [Linear Algebra Appl. 197/198, 143-176 (1994; Zbl 0802.05053)] concerning majorizations between degree sequence and Laplacian eigenvalue sequence of a graph are considered. The conjectures are proved for some special classes of graphs (e.g., threshold graphs, trees, regular graphs).
Reviewer: Dragos Cvetković (Beograd)
MSC:
| 05C50 | Graphs and linear algebra (matrices, eigenvalues, etc.) |
| 05C07 | Vertex degrees |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
Citations:
Zbl 0802.05053References:
| [1] | Merris, R., Laplacian matrices of graphs: a survey, Linear Algebra and its Applications, 1994, 197–198:143–176. · Zbl 0802.05053 · doi:10.1016/0024-3795(94)90486-3 |
| [2] | Faria, I., Permanental roots and the star degree of a graph, Linear Algebra and its Applications, 1985, 64:255–265. · Zbl 0559.05041 · doi:10.1016/0024-3795(85)90281-2 |
| [3] | Chung, F.R.K., Eigenvalues of graphs, in Proceedings of the International Congress of Mathematicians, Zurich, Switzerland, 1994, 1333–1342. · Zbl 0842.05060 |
| [4] | Grone, R., Merris, R., The Laplacian spectrum of graph, (I), SIAM J. Discrete Math., 1994, 7(2): 221–229. · Zbl 0795.05092 · doi:10.1137/S0895480191222653 |
| [5] | Merris, R., Degree maximal graphs are integral, Linear Algebra and its Applications, 1994, 199:381–389. · Zbl 0795.05091 · doi:10.1016/0024-3795(94)90361-1 |
| [6] | Fiedler, M., Algebraic connectivity of graphs, Czechoslovak Math. J., 1973, 23(98):298–305. · Zbl 0265.05119 |
| [7] | Grone, R., Merris, R. and Surder, V.S., The Laplacian spectrum of a graph, SIAM J. Matrix Anal. Appl., 1990, 11:218–238. · Zbl 0733.05060 · doi:10.1137/0611016 |
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