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Isospectral Cayley graphs of some finite simple groups. (English) Zbl 1110.05044

This paper concerns isospectral nonisomorphic finite graphs. Two graphs on \(n\) vertices are isospectral if the multisets of eigenvalues of corresponding adjacency matrices are equal. The authors construct isospectral nonisomorphic Cayley graphs of the finite simple groups \(\text{PSL}_d(\mathbb{F}_q)\) for every \(d \geq 5\) (\(d \neq 6\)) and \(q>2\).
The proof is based on infinite dimensional representation theory and the theory of division algebras over global fields. The graphs considered are the 1-skeletons of Cayley complexes or subgraphs of them. The complexes are obtained as quotients of the Bruhat-Tits building associated with the group \(\text{PGL}_d(F)\), where \(F\) is a local field of positive characteristic.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
11F70 Representation-theoretic methods; automorphic representations over local and global fields
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References:

[1] L. Babai, Spectra of Cayley graphs , J. Combin. Theory Ser. B 27 (1979), 180–189. · Zbl 0338.05110 · doi:10.1016/0095-8956(79)90079-0
[2] R. Brooks, Non-Sunada graphs , Ann. Inst. Fourier (Grenoble) 49 (1999), 707–725. · Zbl 0926.58021 · doi:10.5802/aif.1688
[3] D. I. Cartwright and T. Steger, A family of \(\skew2\tildeA_n\)-groups , Israel J. Math. 103 (1998), 125–140. · Zbl 0923.51010 · doi:10.1007/BF02762271
[4] N. Jacobson, Finite-Dimensional Division Algebras over Fields , Springer, Berlin, 1996. · Zbl 0874.16002
[5] F. T. Leighton, Finite common coverings of graphs , J. Combin. Theory Ser. B 33 (1982), 231–238. · Zbl 0488.05033 · doi:10.1016/0095-8956(82)90042-9
[6] A. Lubotzky, “Cayley graphs: Eigenvalues, expanders and random walks” in Surveys in Combinatorics, 1995 (Stirling, Scotland) , London Math. Soc. Lecture Note Ser. 218 , Cambridge Univ. Press, Cambridge, 1995, 155–189. · Zbl 0835.05033
[7] \BIBPapII
[8] \BIBPapI
[9] \BIBPapIII D. W. Morris and J. Morris, personal communication, 2005.
[10] H. Pesce, Quelques applications de la théorie des représentations en géométrie spectrale , Rend. Mat. Appl. (7) 18 (1998), 1–63. · Zbl 0923.58056
[11] R. S. Pierce, Associative Algebras , Grad. Texts in Math. 88 , Springer, New York, 1982.
[12] M. Ronan, Lectures on Buildings , Perspect. Math. 7 , Academic Press, Boston, 1989. · Zbl 0694.51001
[13] J. J. Seidel, Strongly regular graphs of \(L_2\)-type and of triangular type , Nederl. Akad. Wetensch. Proc. Ser. A 29 (1967), 188–196. · Zbl 0161.20802
[14] T. Sunada, Riemannian coverings and isospectral manifolds , Ann. of Math. (2) 121 (1985), 169–186. JSTOR: · Zbl 0585.58047 · doi:10.2307/1971195
[15] J. Tits, Buildings of Spherical Type and Finite BN-Pairs , Lecture Notes in Math. 386 , Springer, Berlin, 1974. · Zbl 0295.20047
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