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Questions about the branch of combinatorics called graph theory (not to be used for questions concerning the graph of a function). This tag can be further specialized via using it in combination with more specialized tags such as extremal-graph-theory, spectral-graph-theory, algebraic-graph-theory, topological-graph-theory, random-graphs, graph-colorings and several others.
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article by Jacques Tits about automorphism group of a locally finite tree
This is true if the action of the automorphism group is edge-transitive (so in particular the tree has to be biregular). This is precisely the statement of Lemma 2.6(vii and viii) of our paper "Simple …
4
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Is a graph with edges between edges and nodes still a graph?
I'm not aware of a name for such an object, but it can easily be modeled by a single vertex-colored directed graph $\Gamma$:
As vertex set of $\Gamma$, take the union of the vertices and edges of you …