Remark 1

In LIE, IV, p. 33, appendix, the notion of a combinatorial graph was defined as a pair $(A,S)$, where S is a set and A a subset of $\mathfrak{P}(S)$ consisting of sets with two elements; the elements of S are called vertices, those of A edges, and two vertices $x$ and $y\in S$ are said to be joined if $\{x, y\}$ is an edge.

To such a combinatorial graph $\Gamma = (A,S)$, one associates a graph G whose set of vertices is S and whose set of arrows $\widetilde{A}$ is the subset of $S^2$ consisting of pairs of joined vertices, the origin mapping and the end mapping coinciding with the first and second projection of $S^2$ onto S, and the involution $f\mapsto \overline{f}$ being given by the restriction to $\widetilde{A}$ of the mapping $(x, y)\mapsto (y, x)$ of $S^2$ into itself. The mapping which associates with an edge $\{f, f\}$ of G the set $\{o(f), t(f)\}$ is a bijection of the set of edges of G onto that of the edges of the combinatorial graph Γ.

Conversely, every graph such that the origin and the end of every arrow are distinct, and such that an arrow is determined by its origin and its end, is of this form.

The reader will verify that the notions of connectedness, tree, and forest for a combinatorial graph coincide with the corresponding notions for the graph associated with it.