In this survey, we give a friendly introduction from a graph theory perspective to the q-state Potts model, an important statistical mechanics tool for analyzing complex systems in which nearest neighbor interactions determine the aggregate behavior of the system. We present the surprising equivalence of the Potts model partition function and one of the most renowned graph invariants, the Tutte polynomial, a relationship that has resulted in a remarkable synergy between the two fields of study. We highlight some of these interconnections, such as computational complexity results that have alternated between the two fields. The Potts model captures the effect of temperature on the system and plays an important role in the study of thermodynamic phase transitions. We discuss the equivalence of the chromatic polynomial and the zero-temperature antiferromagnetic partition function, and how this has led to the study of the complex zeros of these functions. We also briefly describe Monte Carlo simulations commonly used for Potts model analysis of complex systems. The Potts model has applications as widely varied as magnetism, tumor migration, foam behaviors, and social demographics, and we provide a sampling of these that also demonstrates some variations of the Potts model. We conclude with some current areas of investigation that emphasize graph theoretic approaches.
Over the past twenty years, rectangle visibility graphs have generated considerable interest, in part due to their applicability to VLSI chip design. Here we study unit rectangle visibility graphs, with fixed dimension restrictions more closely modeling the constrained dimensions of gates and other circuit components in computer chip applications. A graph G is a unit rectangle visibility graph (URVG) if its vertices can be represented by closed unit squares in the plane with sides parallel to the axes and pairwise disjoint interiors, in such a way
Building a structure using self-assembly of DNA molecules by origami folding requires finding a route for the scaffolding strand through the desired structure. When the target structure is a 1-complex (or the geometric realization of a graph), an optimal route corresponds to an Eulerian circuit through the graph with minimum turning cost. By showing that it leads to a solution to the 3-SAT problem, we prove that the general problem of finding an optimal route for a scaffolding strand for such structures is NP-hard. We then show that the problem may readily be transformed into a Traveling Salesman Problem (TSP), so that machinery that has been developed for the TSP may be applied to find optimal routes for the scaffolding strand in a DNA origami self-assembly process. We give results for a few special cases, showing for example that the problem remains intractable for graphs with maximum degree 8, but is polynomial time for 4-regular plane graphs if the circuit is restricted to following faces. We conclude with some implications of these results for related problems, such as biomolecular computing and mill routing problems.
Self-assembly using DNA origami methods requires determining a route for the scaffolding strand through the targeted structure. Here we provide strategies and software tools for determining optimal routes for reporter or scaffolding strands through graph-like (balland-rod) constructs. The approach applies to complex constructs, for example arbitrary geometric embeddings of graphs rather than surface meshes, lattice subsets, and meshes on higher genus surfaces than spheres. The software notably allows the user the flexibility of specifying ranked preferences for augmenting edges and for the possible configurations of branched junctions. The greater topological complexity of arbitrary graph embeddings and meshes on higher genus surfaces can result in scaffolding strand routes that are knotted in 3 space, so we also present necessary caveats for these settings.
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