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Geometric distance-regular covers. (English) Zbl 0793.05146

Summary: Let \(G\) be a distance-regular graph with valency \(k\) and least eigenvalue \(\tau\). Delsarte proved that a clique in \(G\) has cardinality at most \(1- {k\over \tau}\). We call a distance-regular graph geometric if each edge lies in a unique clique of this cardinality. Any geometric distance- regular graph is the point graph of a partial linear space with the property that the number of points on a line closest to a given point only depends on the distance between the point and the line. We derive some conditions which imply that a distance-regular graph is geometric, and use them to show that the number of antipodal distance-regular graphs with diameter three and given least eigenvalue is finite.

MSC:

05E30 Association schemes, strongly regular graphs
05C50 Graphs and linear algebra (matrices, eigenvalues, etc.)
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