  We consider universal source coding of unlabeled graphs which are commonly
referred to as graphical structures. We adopt an Erd\"{o}s-R\'enyi model to
generate the random graphical structures. We propose a variant of the
previously introduced Type Size code, where type classes are characterized
based on the number of edges of the graphical structures. The proposed scheme
sorts the graphical structures based on the size of their type classes and
assigns binary sequences to them in this order. The $\epsilon$-coding rate of
the Type Size code (up to the third-order term) for compressing graphical
structures is derived.
