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We give combinatorial models for non-spherical, generic, smooth, complex representations of the group $G=\mathrm{PGL}(2,F)$, where $F$ is a non-Archimedean locally compact field. More precisely we carry on studying the graphs ${\left({\tilde{X}}_k\right)}_{k\ge 0}$ defined in a previous work. We show that such representations may be obtained as quotients of the cohomology of a graph ${\tilde{X}}_k$, for a suitable integer $k$, or equivalently as subspaces of the space of discrete harmonic cochains on such a graph. Moreover, for supercuspidal representations, these models...