This paper addresses the ubiquity of remarkable measures on graphs and their applications. In many queueing systems, it is necessary to take into account the compatibility constraints between users, or between supplies and demands, and so on. The stability region of such systems can then be seen as a set of measures on graphs, where the measures under consideration represent the arrival flows to the various classes of users, supplies, demands, etc., and the graph represents the compatibilities between those classes. In this paper, we show that these 'stabilizing' measures can always be easily constructed as a simple function of a family of weights on the edges of the graph. Second, we show that the latter measures always coincide with invariant measures of random walks on the graph under consideration. Some arguments in the proofs rely on the so-called matching rates of specific stochastic matching models. As a by-product of these arguments, we show that, in several cases, the matching rates are independent of the matching policy, that is, the rule for choosing a match between various compatible elements.
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