# Enumerating and indexing many-body intramolecular interactions: a graph theoretic approach

## TL;DR

This paper introduces a graph-based method to systematically identify and index complex molecular interactions in large biomolecules.

## Contribution

A novel recursive graph-theoretic approach for enumerating and indexing many-body intramolecular interactions.

## Key findings

- The line graph transformation enables recursive mapping of n-body interactions to (n+1)-body terms.
- The method provides a complete characterization of 4-body interactions using directed acyclic graphs.
- Pseudo-code and data structures are provided for efficient implementation in large biomolecular systems.

## Abstract

The central idea observes a recursive mapping of \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}n-body intramolecular interactions to \documentclass[12pt]{minimal}
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\begin{document}$$(n+1)$$\end{document}(n+1)-body terms that is consistent with the molecular topology. Iterative application of the line graph transformation is identified as a natural and elegant tool to accomplish the recursion. The procedure readily generalizes to arbitrary \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}n-body potentials. In particular, the method yields a complete characterization of \documentclass[12pt]{minimal}
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\begin{document}$$4$$\end{document}4-body interactions. The hierarchical structure of atomic index lists for each interaction order \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}n is compactly expressed as a directed acyclic graph. A pseudo-code description of the generating algorithm is given. With suitable data structures (e.g., edge lists or adjacency matrices), automatic enumeration and indexing of \documentclass[12pt]{minimal}
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\begin{document}$$n$$\end{document}n-body interactions can be implemented straightforwardly to handle large bio-molecular systems. Explicit examples are discussed, including a chemically relevant effective potential model of taurocholate bile salt.