  Let $F$ be a probability distribution with support on the non-negative
integers. A model is proposed for generating stationary simple graphs on
$\mathbb{Z}$ with degree distribution $F$ and it is shown for this model that
the expected total length of all edges at a given vertex is finite if $F$ has
finite second moment. It is not hard to see that any stationary model for
generating simple graphs on $\mathbb{Z}$ will give infinite mean for the total
edge length per vertex if $F$ does not have finite second moment. Hence, finite
second moment of $F$ is a necessary and sufficient condition for the existence
of a model with finite mean total edge length.
