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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.

##### MSC:
 05E30 Association schemes, strongly regular graphs 05B25 Combinatorial aspects of finite geometries 05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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##### References:
 [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
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