zbMATH — the first resource for mathematics

Second neighbourhoods of strongly regular graphs. (English) Zbl 0764.05100
Authors’ abstract: Some antipodal distance-regular graphs of diameter three arise as the graph induced by the vertices at distance two from a given vertex in a strongly regular graph. We show that if every vertex in a strongly regular graph \(G\) has this property, then \(G\) is the noncollinearity graph of a special type of semipartial geometry. As these semipartial geometries have all been classified, we obtain a list of the antipodal distance-regular graphs of diameter three that can arise in this way.

05E30 Association schemes, strongly regular graphs
05B25 Combinatorial aspects of finite geometries
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
Full Text: DOI
[1] Biggs, N.L., Algebraic graph theory, () · Zbl 0501.05039
[2] Biggs, N.L., Distance-regular graphs with diameter three, Ann. discrete math., 15, 69-80, (1982) · Zbl 0506.05057
[3] Brouwer, A.E., Distance regular graphs of diameter 3 and strongly regular graphs, Discrete math., 49, 101-103, (1984) · Zbl 0538.05024
[4] Brouwer, A.E.; Cohen, A.M.; Neumaier, A., Distance regular graphs, (1987), preliminary version of a book
[5] Brouwer, A.E.; van Lint, J.H., Strongly regular graphs and partial geometries, (), 85-122 · Zbl 0555.05016
[6] Debroey, I.; Thas, J.A., On semipartial geometries, J. combin. theory ser. A, 25, 242-250, (1978) · Zbl 0399.05012
[7] Dembowski, P., Finite geometries, (1968), Springer Berlin · Zbl 0159.50001
[8] Gardiner, A., Antipodal covering graphs, J. combin. theory ser. B, 16, 255-273, (1974) · Zbl 0267.05111
[9] C.D. Godsil and A.D. Hensel, Distance-regular covers of the complete graph, forthcoming. · Zbl 0771.05031
[10] Hall, J.I., On copolar spaces and graphs, Quart. J. math. Oxford ser. 2, 33, 421-449, (1982) · Zbl 0458.05052
[11] Hirschfeld, J.W.P., Projective geometries over finite fields, (1979), Clarendon Press Oxford · Zbl 0418.51002
[12] Payne, S.; Thas, J.A., Finite generalized quadrangles, (1985), Pitman New York · Zbl 0551.05027
[13] Smith, D.H., Primitive and imprimitive graphs, Quart. J. math. Oxford ser. 2, 22, 551-557, (1971) · Zbl 0222.05111
[14] Somma, C., An infinite family of perfect codes in antipodal graphs, Rend. mat. appl., 3, 7, 465-474, (1983) · Zbl 0546.94014
[15] Taylor, D.E., Regular 2-graphs, Proc. London math. soc., 35, 3, 257-274, (1977) · Zbl 0362.05065
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.