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Interval-regular graphs. (English) Zbl 0542.05051
Summary: An interval-regular graph is a connected graph in which, for any two vertices u and v, the number of neighbours of u on all shortest (u,v)- paths, equals d(u,v). It is proved that in an interval-regular graph the shortest (u,v)-paths induce a hypercube of dimension d(u,v), for any two vertices u and v. The products of complete graphs are characterized as interval-regular graphs satisfying some extra conditions. The extended odd graphs are introduced as critical example with respect to the results proved.

MSC:
05C75 Structural characterization of families of graphs
05C35 Extremal problems in graph theory
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