 Graphs, of the oriented kind arising in say automata theory where multiple edges may connect the same two vertices, can be organized as algebras having two unary operations \(s\) and \(t\) satisfying \(s(s(x)) = t(s(x)) = s(x)\) and \(t(t(x)) = s(t(x)) = t(x)\). The free graph on one generator \(x\) has three elements, \(x, s(x)\), and \(t(x)\), constituting respectively an edge and its two endpoints or vertices. In this framework the vertices are the elements satisfying \(s(x) = x\) (and hence \(t(x) = x\) since \(x = s(x) = t(s(x)) = t(x)\)); all other elements constitute edges. The free graph on \(n\) generators consists of \(n\) such edges, all independent. Other graphs arise by identifying elements. There is no point identifying an edge with either another edge or a vertex since that simply absorbs the first edge into the second entity. This leaves only vertices; identifying two vertices yields a single vertex common to two edges, or to the same edge in the identification \(s(x) = t(x)\) creating a self-loop.