On the evaluation of ultra-high-precision frequency ratio measurements: examining closed loops in a graph theory framework

A simple method is proposed for the evaluation of comparative measurements when closed measurement loops are present in the data. Using a graph theory representation of the data a basis for the residual subspace is readily identified that together with loop closure values provide the input to a matrix equation of the same rank as the number of independent loops in the graph. As an example, ultra-high-precision frequency ratio measurements including optical standards are considered and small corrections to the experimental data are derived, yielding a list of consistent frequency ratios for work on precision frequency standards and discussions relating to a possible future redefinition of the SI second.