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Two distance-regular graphs. (English) Zbl 1254.05199

In this paper two families of distance-regular graphs, the subgraph of the dual polar graph of type \(B_{3}(q)\) induced on the vertices far from a fixed point, and the subgraph of the dual polar graph of type \(D_{4}(q)\) induced on the vertices far from a fixed edge are constructed. The first graph is defined as follows: let \(W\) be a vector space of dimension 3 over the field \(F_{q}\), provided with an outer product \(\times\). Let \(Z\) be the graph with vertex set \(W\times W\) where \((u,u')\) is adjacent to \((v,v')\) if and only if \((u,u')\neq (v,v')\) and \(u \times v+u'-v'=0\). Then \(Z\) is distance-regular of diameter 3 on \(q^{6}\) vertices. The latter is the extended bipartite double of the former.

MSC:

05E30 Association schemes, strongly regular graphs
05C12 Distance in graphs
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[1] Blok, R.J., Brouwer, A.E.: The geometry far from a residue. In: di Martino, L., Kantor, W.M., Lunardon, G., Pasini, A., Tamburini, M.C. (eds.) Groups and Geometries, pp. 29–38. Birkhäuser, Basel (1998) · Zbl 0899.51005
[2] Brouwer, A.E.: Additions and corrections to [3]. http://www.win.tue.nl/\(\sim\)aeb/drg/index.html
[3] Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Heidelberg (1989) · Zbl 0747.05073
[4] Pasini, A.: Diagram Geometries. Oxford University Press, Oxford (1994) · Zbl 0813.51002
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