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Let us call a simple graph on n ≥ 2 vertices a prime gap graph if its vertex degrees are 1 and the first n - 1 prime gaps. We show that such a graph exists for every large n, and in fact for every n ≥ 2 if we assume the Riemann hypothesis. Moreover, an infinite sequence of prime gap graphs can be generated by the so-called degree preserving growth process. This is the first time a naturally occurring infinite sequence of positive integers is identified as graphic. That is, we show the existence of an interesting, and so far unique, infinite combinatorial object.