On distance-regular Cayley graphs on abelian groups. (English) Zbl 1297.05112
Summary: Let \(G\) denote a finite abelian group with identity 1 and let \(S\) denote an inverse-closed subset of \(G \smallsetminus \{1 \}\), which generates \(G\) and for which there exists \(s \in S\), such that \(\langle S \smallsetminus \{s, s^{- 1} \} \rangle \neq G\). In this paper we obtain the complete classification of distance-regular Cayley graphs \(\operatorname{Cay}(G; S)\) for such pairs of \(G\) and \(S\).
References:
| [1] | Bridges, W. G.; Mena, R. A., Rational circulants with rational spectra and cyclic strongly regular graphs, Ars Combin., 8, 143-161 (1979) · Zbl 0443.05046 |
| [2] | Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-Regular Graphs (1998), Springer-Verlag: Springer-Verlag New York |
| [3] | Leifman, Y.; Muzychuk, M., Strongly regular Cayley graphs over the group \(Z_{p^n} \oplus Z_{p^n}\), Discrete Math., 305, 219-239 (2005) · Zbl 1078.05038 |
| [4] | Ma, S. L., Partial difference sets, Discrete Math., 52, 75-89 (1984) · Zbl 0548.05012 |
| [5] | Ma, S. L., A survey of partial difference sets, Des. Codes Cryptogr., 4, 221-261 (1994) · Zbl 0798.05008 |
| [6] | Marušič, D., Strong regularity and circulant graphs, Discrete Math., 78, 119-125 (1989) · Zbl 0689.05026 |
| [7] | Miklavič, Š.; Potočnik, P., Distance-regular circulants, European J. Combin., 24, 777-784 (2003) · Zbl 1026.05107 |
| [8] | Miklavič, Š.; Potočnik, P., Distance-regular Cayley graphs on dihedral groups, J. Combin. Theory Ser. B, 97, 14-33 (2007) · Zbl 1107.05102 |
| [9] | Shrikhande, S. S., The uniqueness of the \(L_2\) association scheme, Ann. Math. Stat., 30, 781-798 (1959) · Zbl 0086.34802 |
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