  Pairs of graded graphs, together with the Fomin property of graded graph
duality, are rich combinatorial structures providing among other a framework
for enumeration. The prototypical example is the one of the Young graded graph
of integer partitions, allowing us to connect number of standard Young tableaux
and numbers of permutations. Here, we use operads, that algebraic devices
abstracting the notion of composition of combinatorial objects, to build pairs
of graded graphs. For this, we first construct a pair of graded graphs where
vertices are syntax trees, the elements of free nonsymmetric operads. This pair
of graphs is dual for a new notion of duality called $\phi$-diagonal duality,
similar to the ones introduced by Fomin. We also provide a general way to build
pairs of graded graphs from operads, wherein underlying posets are analogous to
the Young lattice. Some examples of operads leading to new pairs of graded
graphs involving integer compositions, Motzkin paths, and $m$-trees are
considered.
