  Let us call a simple graph on $n\geq 2$ 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 $n\geq 2$ 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.
