In 1971, M. M. Peixoto [15] introduced an importanttopologicalinvariant of Morse–Smale flows on surfaces, which he called adistinguishedgraph $X^*$ associated with a given flow.Here we show how the Peixoto invariant can be essentially simplifiedand revised by adopting a purely topological point of viewconnected with the embeddings of arbitrary graphs into compact surfaces.The newly obtained invariant, $X^R$, isa rotation of a graph $X$ generated by a Morse–Smale flow. (A rotation$R$ is acyclic order of edges given in every vertex of $X$.)The invariant $X^R$ ‘reads-off’ the topological informationcarried by a flow,being in a one-to-one correspondence with the topological equivalence classesof Morse–Smale flowsAnd foliations, see[3]. We do not treatthe case of foliations, bearing in mind that they are defined by involutiveflows on covering manifolds [9]..As a counterpart to the equivalence result we prove a realization theorem foran ‘abstractly given’ $X^R$. (Our methodsare completely different from those of Peixoto and theyclarify the connections between graphs and flows on surfaces.)The idea of ‘rotation systems’ on graphs can be further exploitedin thestudy of recurrent flows (and foliations) with several disjoint quasiminimalsets on surfaces [10].