An algebraic approach to enumerating non-equivalent double traces in graphs

TL;DR

This paper presents an algebraic method and a branch-and-bound algorithm for enumerating non-equivalent double traces in graphs, which are crucial for designing biomolecular polyhedra in nanotechnology.

Contribution

It introduces a novel algebraic approach to analyze automorphisms of double traces and develops an efficient enumeration algorithm for these structures.

Findings

The algebraic approach effectively characterizes automorphisms of double traces.

The branch-and-bound algorithm can enumerate all non-equivalent double traces.

Application potential in designing biomolecular polyhedra.

Abstract

Recently designed biomolecular approaches to build single chain polypeptide polyhedra as molecular origami nanostructures have risen high interest in various double traces of the underlying graphs of these polyhedra. Double traces are walks that traverse every edge of the graph twice, usually with some additional conditions on traversal direction and vertex neighborhood coverage. Given that double trace properties are intimately related to theefficiency of polypeptide polyhedron construction, enumerating all different possible double traces and analyzing their properties is an important step in the construction. In the paper, we study the automorphism group of double traces and present an algebraic approach to this problem, yielding a branch-and-bound algorithm.