A pairing formula for resonant states on finite regular graphs

On a finite regular graph, (co)resonant states are eigendistributions of the transfer operator associated to the shift on one-sided infinite non-backtracking paths. We introduce two pairings of resonant and coresonant states, the vertex pairing which involves only the dependence on the initial/terminal vertex of the path, and the geodesic pairing which is given by integrating over all geodesics the evaluation of the coresonant state on the first half of the geodesic times the resonant state on the second half. The main result is that these two pairings coincide up to a constant which depends on the resonance, i.e. the corresponding eigenvalue of the transfer operator.