  A compact symplectic manifold $(M, \omega)$ is said to satisfy the
hard-Lefschetz condition if it is possible to develop an analogue of Hodge
theory for $(M, \omega)$. This loosely means that there is a notion of
harmonicity of differential forms in $M$, depending on $\omega$ alone, such
that every de Rham cohomology class in has a $\omega$-harmonic representative.
In this article, we study two non-equivalent families of diagonal
almost-abelian Lie algebras that admit a distinguished almost-K\"ahler
structure and compute their cohomology explicitly. We show that they satisfy
the hard-Lefschetz condition with respect to any left-invariant symplectic
structure by exploiting an unforeseen connection with Kneser graphs. We also
show that for some choice of parameters their associated simply connected,
completely solvable Lie groups admit lattices, thereby constructing examples of
almost-K\"ahler solvmanifolds satisfying the hard-Lefschetz condition, in such
a way that their de Rham cohomology is fully known.
