Arc-transitive antipodal distance-regular covers of complete graphs related to \(\mathrm{SU}_3(q)\). (English) Zbl 1351.05113
Summary: In this paper, we classify antipodal distance-regular graphs of diameter three that admit an arc-transitive action of \(\mathrm{SU}_3(q)\). In particular, we find a new infinite family of distance-regular antipodal \(r\)-covers of a complete graph on \(q^3 + 1\) vertices, where \(q\) is odd and \(r\) is any divisor of \(q + 1\) such that \((q + 1) / r\) is odd. Further, we find several new constructions of arc-transitive antipodal distance-regular graphs of diameter three in case \(\lambda = \mu\).
MSC:
| 05C25 | Graphs and abstract algebra (groups, rings, fields, etc.) |
| 05C12 | Distance in graphs |
| 05C70 | Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) |
Keywords:
antipodal graph; arc-transitive graph; automorphism group; distance-regular graph; covering graphSoftware:
GAPReferences:
| [1] | Aschbacher, M., Finite Group Theory (2000), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0965.20009 |
| [2] | Brouwer, A. E.; Cohen, A. M.; Neumaier, A., Distance-regular Graphs, 494 (1989), Springer-Verlag: Springer-Verlag Berlin · Zbl 0747.05073 |
| [4] | Cameron, P. J., Covers of graphs and EGQs, Discrete Math., 97, 1-3, 83-92 (1991) · Zbl 0763.05077 |
| [5] | Devillers, A.; Giudici, M., Involution graphs where the product of two adjacent vertices has order three, J. Aust. Math. Soc., 85, 305-322 (2008) · Zbl 1177.20022 |
| [6] | Dixon, J. D.; Mortimer, B., Permutation Groups (1996), Springer: Springer New York · Zbl 0951.20001 |
| [7] | Godsil, C. D., Krein covers of complete graphs, Australas. J. Combin., 6, 245-255 (1992) · Zbl 0776.05086 |
| [8] | Godsil, C. D.; Hensel, A. D., Distance regular covers of the complete graph, J. Combin. Theory Ser. B., 56, 205-238 (1992) · Zbl 0771.05031 |
| [9] | Godsil, C. D.; Liebler, R. A.; Praeger, C. E., Antipodal distance transitive covers of complete graphs, European J. Combin., 19, 4, 455-478 (1998) · Zbl 0914.05035 |
| [10] | Klin, M.; Pech, Ch., A new construction of antipodal distance-regular covers of complete graphs through the use of Godsil-Hensel matrices, Ars Math. Contemp., 4, 205-243 (2011) · Zbl 1245.05136 |
| [11] | Lorimer, P., Vertex-transitive graphs: symmetric graphs of prime valency, J. Graph Theory, 8, 55-69 (1984) · Zbl 0535.05031 |
| [12] | Makhnev, A. A.; Paduchikh, D. V.; Tsiovkina, L. Yu., Arc-transitive distance-regular covers of cliques with \(\lambda = \mu \), Proc. Stekl. Inst. Math., 284, Suppl.1, 124-134 (2014) · Zbl 1304.05038 |
| [13] | Makhnev, A. A.; Paduchikh, D. V.; Tsiovkina, L. Yu., Edge-symmetric distance-regular coverings of cliques: the affine case, Sib. Math. J., 54, 6, 1353-1367 (2013) · Zbl 1282.05065 |
| [14] | Mukhamet’yanov, I. T., Graphs on the class of conjugate involutions of the group \(L_2(2^m)\), Nauch. Obozren. (5), 133-139 (2013), (in Russian) |
| [15] | Suzuki, M., On a class of doubly transitive groups, Ann. Math., Second Series, 75, 1, 105-145 (1962) · Zbl 0106.24702 |
| [16] | Suzuki, M., On a class of doubly transitive groups II, Ann. Math., Second Series, 79, 3, 514-589 (1964) · Zbl 0123.25101 |
| [17] | Suzuki, M., Finite groups of even order in which Sylow 2-groups are independent, Ann. Math., Second Series, 80, 1, 58-77 (1964) · Zbl 0122.03202 |
| [18] | Suzuki, M., A characterization of the 3-dimensional projective unitary group over a finite field of odd characteristic, J. Algebra (2), 1-14 (1965) · Zbl 0134.03102 |
| [19] | Taylor, D. E., Two-graphs and doubly transitive groups, J. Combin. Theory Ser. A., 61, 113-122 (1992) · Zbl 0760.05056 |
| [21] | Tsiovkina, L. Yu., Two new infinite families of arc-transitive antipodal distance-regular graphs of diameter three with \(\lambda = \mu\) related to groups \(S z(q)\) and \({}^2 G_2(q)\), J. Algebraic Combin., 41, 1079-1087 (2015) · Zbl 1333.05320 |
| [22] | Tsiovkina, L. Yu., On affine distance-regular covers of complete graphs, Sib. Electr. Math. Reports, 12, 998-1005 (2015), (in Russian) · Zbl 1341.05125 |
| [24] | Ward, H. N., On Ree’s series of simple groups, Trans. Amer. Math. Soc., 121, 1, 62-89 (1966) · Zbl 0139.24902 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
