A Representation of Virial Coefficients That Avoids the Asymptotic Catastrophe

We investigate the notion of an asymptotic catastrophe in representations of Mayer coefficients. The manifestations of the catastrophe and its formal definition are given. The significance of the definition introduced for an asymptotic catastrophe is clarified. Virial-coefficient representations that are free of the asymptotic catastrophe phenomenon are given. Sets of labeled graphs (blocks) nonseparable in the Harary sense are expanded into classes labeled by cycle ensembles satisfying specific conditions, and the representations are based on these expansions. These cycle ensembles are called frame cycle ensembles. The same classes can be labeled by special blocks, which are called frames. The frames are brought into one-to-one correspondence with the frame cycle ensembles. In the block classification, frames play a role similar to the role of trees in the tree classification of connected labeled graphs. A tree classification of frame cycle ensembles is introduced. We prove that the described virial-coefficient representations are free of the asymptotic catastrophe phenomenon.