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The attractiveness of the flow-graph analysis of systems lies to a great extent in the ability to manipulate the graph by inspection. Nevertheless, in large flow graphs the inspection technique is often a shaky one for writing transmission equations. If one insists on using the flow-graph approach, a systematic mathematical means is needed to assure that no loops, paths, or contiguous conditions among loops and/or paths are missed. This paper describes an analytical-combinatorial method, which produces error-free transmission equations. Use is made of suitably modified vertex matrices and combinatorially developed path-vertex-loop matrices.