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The girth of a directed distance-regular graph. (English) Zbl 0733.05044
Summary: Let \(G=(V,E)\) be a connected digraph with the usual (non-symmetric) metric \(\partial\). For u,v\(\in V\), let \[ p_{ij}(u,v)=\#\{w\in V:\;\partial (u,w)=i,\quad \partial (w,v)=j\}. \] G is said to be distance- regular if \(\partial (u,v)=\partial (u',v')\) implies \(p_{i,j}(u,v)=p_{i,j}(u',v')\) for all i, j. In this article, it is shown that if G is a directed distance-regular graph (other than a directed cycle or its coclique extension), then G has girth \(g\leq 8\).

MSC:
05C20 Directed graphs (digraphs), tournaments
05C38 Paths and cycles
05C12 Distance in graphs
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