The Number of Ways to Assemble a Graph

TL;DR

This paper introduces the concept of assembly trees for graphs, providing formulas and asymptotic analysis for their enumeration, especially for bipartite and tripartite graphs, with applications to macromolecular assembly.

Contribution

It defines assembly trees for graphs and derives explicit formulas, generating functions, recurrence relations, and asymptotic formulas for their counts, advancing understanding of graph assembly processes.

Findings

Explicit formulas for assembly trees of certain graph families

Recurrence relations for counting assembly trees of bipartite and tripartite graphs

Asymptotic formulas for the number of assembly trees in specific graph classes

Abstract

Motivated by the question of how macromolecules assemble, the notion of an {\it assembly tree} of a graph is introduced. Given a graph $G$, the paper is concerned with enumerating the number of assembly trees of $G$, a problem that applies to the macromolecular assembly problem. Explicit formulas or generating functions are provided for the number of assembly trees of several families of graphs, in particular for what we call $(H,\phi)$-graphs. In some natural special cases, we apply powerful recent results of Zeilberger and Apagodu on multivariate generating functions, and results of Wimp and Zeilberger, to deduce recurrence relations and very precise asymptotic formulas for the number of assembly trees of the complete bipartite graphs $K_{n,n}$ and the complete tripartite graphs $K_{n,n,n}$. Future directions for reseach, as well as open questions, are suggested.