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We study the duals of a certain class of finite-dimensional operator systems,namely the class of operator systems associated to tolerance relations onfinite sets or equivalently the class of operator systems that are associatedwith graphs. In the case where the graphs associated with these operatorsystems are chordal we are able to find concrete realizations of their duals assitting inside of finite-dimensional $C^*$-algebras. We then use these concreterealizations to compute the $C^*$-envelopes, propagation numbers and extremalrays of these duals in the chordal case. Finally, we exemplify our results byapplying them to operator systems of band matrices.

Operator systems for tolerance relations on finite sets