Non-Sunada graphs

Robert Brooks

Annales de l'institut Fourier (1999)

  • Volume: 49, Issue: 2, page 707-725
  • ISSN: 0373-0956

Abstract

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We consider the question of whether there is a converse to the Sunada Theorem in the context of k -regular graphs. We give a weak converse to the Sunada Theorem, which gives a necessary and sufficient condition for two graphs to be isospectral in terms of a Sunada-like condition, and show by example that a strong converse does not hold.

How to cite

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Brooks, Robert. "Non-Sunada graphs." Annales de l'institut Fourier 49.2 (1999): 707-725. <http://eudml.org/doc/75352>.

@article{Brooks1999,
abstract = {We consider the question of whether there is a converse to the Sunada Theorem in the context of $k$-regular graphs. We give a weak converse to the Sunada Theorem, which gives a necessary and sufficient condition for two graphs to be isospectral in terms of a Sunada-like condition, and show by example that a strong converse does not hold.},
author = {Brooks, Robert},
journal = {Annales de l'institut Fourier},
keywords = {-regular graphs; Laplacian; isospectrality},
language = {eng},
number = {2},
pages = {707-725},
publisher = {Association des Annales de l'Institut Fourier},
title = {Non-Sunada graphs},
url = {http://eudml.org/doc/75352},
volume = {49},
year = {1999},
}

TY - JOUR
AU - Brooks, Robert
TI - Non-Sunada graphs
JO - Annales de l'institut Fourier
PY - 1999
PB - Association des Annales de l'Institut Fourier
VL - 49
IS - 2
SP - 707
EP - 725
AB - We consider the question of whether there is a converse to the Sunada Theorem in the context of $k$-regular graphs. We give a weak converse to the Sunada Theorem, which gives a necessary and sufficient condition for two graphs to be isospectral in terms of a Sunada-like condition, and show by example that a strong converse does not hold.
LA - eng
KW - -regular graphs; Laplacian; isospectrality
UR - http://eudml.org/doc/75352
ER -

References

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  3. [BGG] R. BROOKS, R. GORNET, W. GUSTAFSON, Mutually Isospectral Riemann Surfaces, Adv. Math., 138 (1998), 306-322. Zbl0997.53031MR99k:58184
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  5. [CDGT] D. CVETKOVIC, M. DOOB, I. GUTMAN, A. TARGASEV, Recent Results in the Theory of Graph Spectra, Ann. Disc. Math. 36, North Holland, 1988. Zbl0634.05054MR89d:05130
  6. [GGSWW] C. GORDON, R. GORNET, D. SCHUETH, D. WEBB, E. WILSON, Isospectral Deformations of Closed Riemannian Manifolds with Different Scalar Curvature, Ann. Inst. Fourier, 48-2 (1998), 593-607. Zbl0922.58083MR99b:53049
  7. [Le] F. LEIGHTON, Finite Common Coverings of Graphs, J. Comb. Theory, B 33 (1982), 231-238. Zbl0488.05033MR85a:05068
  8. [LMZ] A. LUBOTZKY, S. MOZES, R. ZIMMER, Superrigidity for the Commensurability Group of Tree Lattices, Comm. Math. Helv., 69 (1994), 523-548. Zbl0839.22011MR96a:20032
  9. [Pe1] H. PESCE, Quelques applications de la théorie des représentations en géométrie spectrale, Thèse d'habilitation, Grenoble, 1997; Rendiconti di Mathematica, 18 (1998), 1-64. Zbl0923.58056MR99j:58215
  10. [Pe2] H. PESCE, Une réciproque générique du théorème de Sunada, Compositio Math., 109 (1997), 357-365. Zbl0889.58080MR98k:58232
  11. [Pe3] H. PESCE, Variétés isospectrales et représentations de groupes, in Brooks, Gordon, and Perry ed., Geometry of the Spectrum, Contemp. Math, 173 (1994), 231-240. Zbl0814.58041MR95k:58169
  12. [Qu] G. QUENELL, The Combinatorics of Seidel Switching, preprint. 
  13. [Su] T. SUNADA, Riemannian Coverings and Isospectral Manifolds, Ann. Math., 121 (1985), 169-186. Zbl0585.58047MR86h:58141

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