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Show that a Cayley digraph is strongly connected if and only if it is weakly connected. (A digraph is strongly connected if there is a directed path between any two vertices. It is weakly connected if the underlying graph is connected.) Hint: this is true for any regular digraph.

Joel.Kh
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  • This holds in any graph where every vertex has the same in- and out-degrees, see: https://math.stackexchange.com/a/2506630/81032 – Dániel Garamvölgyi Sep 22 '20 at 17:20
  • @DánielG., Sorry I'm a little bit new with Cayley graphs and Graph theory, You mean that in every Cayley graph which is weakly connected, every vertex has the same in- and out-degrees? – Joel.Kh Sep 22 '20 at 18:07
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    A Cayley graph (with the usual definition where its given by a group and a generating set) is always regular, and in particular it has the same in- and out-degrees. Moreover, it is always weakly (and thus strongly) connected, so your problem seems a bit peculiar. Maybe you are using a different definition of a Cayley graph? – Dániel Garamvölgyi Sep 22 '20 at 20:27

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