A Prüfer-Sequence Based Algorithm for Calculating the Size of Ideal Randomly Branched Polymers

Branched polymers can be represented as tree graphs. A one-to-one correspondence exists between a tree graph comprised of N labeled vertices and a sequence of N - 2 integers, known as the Prüfer sequence. Permutations of this sequence yield sequences corresponding to tree graphs with the same vertex-degree distribution but (generally) different branching patterns. Repeatedly shuffling the Prüfer sequence we have generated large ensembles of random tree graphs, all with the same degree distributions. We also present and apply an efficient algorithm to determine graph distances directly from their Prüfer sequences. From the (Prüfer sequence derived) graph distances, 3D size metrics, e.g., the polymer's radius of gyration, R_g, and average end-to-end distance, were then calculated using several different theoretical approaches. Applying our method to ideal randomly branched polymers of different vertex-degree distributions, all their 3D size measures are found to obey the usual N^1/4 scaling law. Among the branched polymers analyzed are RNA molecules comprised of equal proportions of the four - randomly distributed - nucleotides. Prior to Prüfer shuffling, the vertices of their representative tree graphs, these "random-sequence" RNAs exhibit an R_g ∼ N^1/3 scaling.