Abstract
A Frequency Assignment Problem (FAP) is the problem that arises when frequencies have to be assigned to a given set of transmitters so that spectrum is used efficiently and the interference between the transmitters is minimal. In this paper we see the frequency assignment problem as a generalised graph colouring problem, where transmitters are presented by vertices and interaction between two transmitters by a weighted edge. We generalise some properties of Laplacian matrices that hold for simple graphs. We investigate the use of Laplacian eigenvalues and eigenvectors as tools in the analysis of properties of a FAP and its generalised chromatic number (the so-called span).
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References
W.N. Anderson and T.D. Morley, Eigenvalues of the Laplacian of a graph, Linear and Multilinear Algebra 18 (1985) 141–145.
N. Biggs, How to compute the spectral density of a lattice and its quotients, CDAM Report Series, LSE (1994).
R. Grone and G. Zimmermann, Large eigenvalues of the Laplacian, Linear and Multilinear Algebra 28 (1990) 45–47.
R.A. Horn and C.H. Johnson, Matrix Analysis (Cambridge University Press, Cambridge, 1985).
R. Merris, A note on Laplacian graph eigenvalues, Linear Algebra and its Applications 285 (1998) 33–35.
B. Mohar, Some applications of Laplace eigenvalues of graphs, in: Graph Symmetry: Algebraic Methods and Applications, eds. G. Hahn and G. Sabidussi, NATO Advanced Science Institutes Series C 497 (Kluwer Academic, 1997) 225-275.
J. van den Heuvel and S. Pejiç, Using Laplacian eigenvalues and eigenvectors in the analysis of frequency assignment problems, CDAM Report Series, LSE (2000).
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van den Heuvel, J., Pejić, S. Using Laplacian Eigenvalues and Eigenvectors in the Analysis of Frequency Assignment Problems. Annals of Operations Research 107, 349–368 (2001). https://doi.org/10.1023/A:1014927805247
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DOI: https://doi.org/10.1023/A:1014927805247