Geodesic
On the Fastest Moving Off from a Vertex in Directed Regular Graphs
Matematičeskie zametki, Tome 82 (2007) no. 5, pp. 770-782.

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Let $\Gamma$ be a directed regular locally finite graph, and let $\overline\Gamma$ be the undirected graph obtained by forgetting the orientation of $\Gamma$. Let $x$ be a vertex of $\Gamma$ and let $n$ be a nonnegative integer. We study the length of the shortest directed path in $\Gamma$ starting at $x$ and ending outside of the ball of radius $n$ centered at $x$ in $\overline\Gamma$.
Keywords: directed graph, undirected graph, locally finite graph
Mots-clés : automorphism group. 
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     author = {V. I. Trofimov},
     title = {On the {Fastest} {Moving} {Off} from a {Vertex} in {Directed} {Regular} {Graphs}},
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V. I. Trofimov. On the Fastest Moving Off from a Vertex in Directed Regular Graphs. Matematičeskie zametki, Tome 82 (2007) no. 5, pp. 770-782. http://geodesic.mathdoc.fr/item/MZM_2007_82_5_a10/

[1] V. I. Trofimov, “On geometric properties of directed vertex-symmetric graphs”, European J. Combin., 27:5 (2006), 690–700 | DOI | MR | Zbl