This paper considers the following questions: (1) what is the relationship between the method of paired comparisons and Rasch measurement theory? (2) what is the relationship between the method of paired comparisons and graph theory? and (3) what can graph theory contribute to the understanding of Rasch measurement theory? It is specifically shown how the method of paired comparisons can lead to the Rasch model, just as consideration of the Rasch model can lead to a pairwise algorithm for estimating the parameters of the Rasch model. Furthermore, both graph theory and previously unexplored aspects of the method of paired comparisons are used to increase understanding and utility of a pairwise algorithm for estimating parameters of the Rasch model as presented by B. Choppin (1985). Bringing together these three lines of inquiry enhances understanding of the Rasch model and provides more effective means of analysis. Appendixes present the conversion of the cumulative distribution function and Statistical Analysis System routines for the analysis. (Contains 8 figures, 7 tables, and 66 references.) (Author/SLD)