Non-Sunada graphs
Annales de l'Institut Fourier, Volume 49 (1999) no. 2, p. 707-725

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.

Nous considérons la question de l’existence d’une réciproque du théorème de Sunada dans le cadre des graphes k-réguliers. Nous étudions une réciproque faible du théorème de Sunada qui donne une condition nécessaire et suffisante pour que deux graphes soient isospectraux, en termes d’une condition “presque-Sunada”, et proposons un contre-exemple qui montre qu’il n’y a pas de réciproque forte.

@article{AIF_1999__49_2_707_0,
     author = {Brooks, Robert},
     title = {Non-Sunada graphs},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {49},
     number = {2},
     year = {1999},
     pages = {707-725},
     doi = {10.5802/aif.1688},
     zbl = {0926.58021},
     mrnumber = {2000i:58062},
     language = {en},
     url = {http://www.numdam.org/item/AIF_1999__49_2_707_0}
}
Brooks, Robert. Non-Sunada graphs. Annales de l'Institut Fourier, Volume 49 (1999) no. 2, pp. 707-725. doi : 10.5802/aif.1688. http://www.numdam.org/item/AIF_1999__49_2_707_0/

[AG] D. Angluin, A. Gardiner, Finite Common Coverings of Pairs of Graphs, J. Comb. Theory, B 30 (1981), 184-187. | MR 82g:05071 | Zbl 0426.05044

[Br] R. Brooks, Twist Surfaces, to appear in Proc. Cortona Conf. | Zbl 0947.30032

[BGG] R. Brooks, R. Gornet, W. Gustafson, Mutually Isospectral Riemann Surfaces, Adv. Math., 138 (1998), 306-322. | MR 99k:58184 | Zbl 0997.53031 | Zbl 01228411

[BPP] R. Brooks, P. Petersen, P. Perry, On Cheeger's Inequality, Comm. Math. Helv., 68 (1993), 599-621. | MR 94k:58150 | Zbl 0809.53049

[CDGT] D. Cvetkovic, M. Doob, I. Gutman, A. Targasev, Recent Results in the Theory of Graph Spectra, Ann. Disc. Math. 36, North Holland, 1988. | MR 89d:05130 | Zbl 0634.05054

[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. | Numdam | MR 99b:53049 | Zbl 0922.58083

[Le] F. Leighton, Finite Common Coverings of Graphs, J. Comb. Theory, B 33 (1982), 231-238. | MR 85a:05068 | Zbl 0488.05033 | Zbl 0502.05020

[LMZ] A. Lubotzky, S. Mozes, R. Zimmer, Superrigidity for the Commensurability Group of Tree Lattices, Comm. Math. Helv., 69 (1994), 523-548. | MR 96a:20032 | Zbl 0839.22011

[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. | MR 99j:58215 | Zbl 0923.58056

[Pe2] H. Pesce, Une réciproque générique du théorème de Sunada, Compositio Math., 109 (1997), 357-365. | MR 98k:58232 | Zbl 0889.58080

[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. | MR 95k:58169 | Zbl 0814.58041

[Qu] G. Quenell, The Combinatorics of Seidel Switching, preprint.

[Su] T. Sunada, Riemannian Coverings and Isospectral Manifolds, Ann. Math., 121 (1985), 169-186. | MR 86h:58141 | Zbl 0585.58047