Abstract
Any directed graph G with N vertices and J edges has an associated line-graph L(G) where the J edges form the vertices of L(G). We show that the non-zero eigenvalues of the adjacency matrices are the same for all graphs of such a family L n(G). We give necessary and sufficient conditions for a line-graph to be quantisable and demonstrate that the spectra of associated quantum propagators follow the predictions of random matrices under very general conditions. Line-graphs may therefore serve as models to study the semiclassical limit (of large matrix size) of a quantum dynamics on graphs with fixed classical behaviour.
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T. Kottos and U. Smilansky, Phys. Rev. Lett. 79:4794(1997).
T. Kottos and U. Smilansky, Ann. Phys. NY 274:76(1999).
G. Berkolaiko and J. P. Keating, J. Phys. A: Math. Gen. 32:7814(1999).
H. Schanz and U. Smilansky, Philos. Mag. B 80:1999(2000).
H. Schanz and U. Smilansky, Phys. Rev. Lett. 84:1427(2000).
T. Kottos and U. Smilansky, Phys. Rev. Lett. 85:968(2000).
F. Barra and P. Gaspard, J. Statist. Phys. 101:283(2000).
G. Berkolaiko, E. B. Bogomolny, and J. P. Keating J. Phys. A: Math. Gen. 34:335(2001).
F. Barra and P. Gaspard, Phys. Rev. E 63:066215(2001).
G. Berkolaiko, H. Schanz, and R. S. Whitney, Phys. Rev. Lett. 88:104101(2002).
G. Tanner, J. Phys. A 35:5985(2002).
G. Tanner, J. Phys. A: Math. Gen. 33:3567(2000).
G. Tanner, J. Phys. A: Math. Gen. 34:8485(2001).
P. Pakoński, K. Życzkowski, and M. Kuś, J. Phys. A: Math. Gen. 34:9303(2001).
D. M. Cvetković, M. Doob, and H. Sachs, Spectra of Graphs: Theory and Application, 2nd ed. (Deutscher Verlag der Wissenschaften, Berlin, 1982).
J. Bang-Jensen and G. Gutin, Digraphs: Theory, Algorithms and Applications (Springer, London, 2001).
A. Mostowski and M. Stark, Introduction to Higher Algebra (Pergamon, Oxford, 1964).
A. Katok and B. Hasselblatt, Introduction to the Modern Theory of Dynamical Systems (Cambridge University Press, Cambridge, 1995).
A. W. Marshall and I. Olkin, The Theory of Majorizations and Its Applications (Academic Press, New York, 1979).
G. Berkolaiko, J. Phys. A: Math. Gen. 34:L319(2001).
K. Życzkowski, W. Słomczyński, M. Kuś, and H.-J. Sommers, preprint Random Unistochastic Matrices nlin.CD/0112036 (2001).
S. Severini,preprintOn the Pattern of Unitary Matricesmath.CO/02051872002.
P. I. Richards, SIAM Rev. 9:548(1967).
R. L. Hemminger and L. W. Beineke, in Selected Topics in Graph Theory, L. W. Beineke and R. J. Wilson, eds. (Academic Press, London, 1978), p. 271.
F. J. Dyson, J. Math. Phys. 3:140, 157, 166 (1962).
F. Haake, Quantum Signatures of Chaos, 2nd Ed. (Springer-Verlag, Berlin, 2000).
M. L. Mehta, Random Matrices, 2nd Ed. (Academic Press, New York, 1991).
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Pakoński, P., Tanner, G. & Życzkowski, K. Families of Line-Graphs and Their Quantization. Journal of Statistical Physics 111, 1331–1352 (2003). https://doi.org/10.1023/A:1023012502046
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DOI: https://doi.org/10.1023/A:1023012502046