Applied Combinatorial Mathematics

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Cyclic Designs

This paper is a combinatorial and computational study of the moduli space of tropical curves of genus g, the moduli space of principally polarized tropical abelian varieties, and the tropical Torelli map. These objects were introduced recently by Brannetti, Melo, and Viviani. Here, we give a new definition of the category of stacky fans, of which the aforementioned moduli spaces are objects and the Torelli map is a morphism. We compute the poset of cells of tropical M_g and of the tropical Schottky locus for genus at most 5. We show that tropical A_g is Hausdorff, and we also construct a finite-index cover for A_3 which satisfies a tropical-type balancing condition. Many different combinatorial objects, including regular matroids, positive semidefinite forms, and metric graphs, play a role.

/pdf/combinatorics-of-the-tropical-torelli-map-1t37gmbhtv.pdf

Combinatorics of the tropical Torelli map

The equivalence of the perimeter topological equations derived by Dias and the 13 possible modes of hexagon adjacency in fused benzenoid systems subsequently presented by Cyvin, Gutman, and collaborators is shown. The aufbau principle for generation of all fused total resonant sextet (TRS) benzenoid hydrocarbons is proved. The precise distinction and topological properties between fused, nonfused, strain-free, strained, helical, and nonhelical benzenoid hydrocarbons are carefully delineated. The TRS benzenoid isomers have the maximum number of bay regions, and expressions for the number of bay regions that any given TRS benzenoid will have are given.

Perimeter topology of benzenoid polycyclic hydrocarbons.

The “configurational” approach is shown to shift several stereochemical problems to the analysis of combinatorial rather than geometrical objects. On the basis of this approach, algebraic chirality criteria for point 3D configurations and their superpositions (labeled, graph, and molecular configurations) are deduced. The application of the suggested criteria makes it possible to elaborate a new classification system which is directly applied to rigid chiral objects (i.e., to fixed spatial realizations of complete and hydrogen-depleted molecular graphs and to skeleton, framework, and core graphs as well). Several examples of rigid F-, G-, N-, P-, and S-chiral molecules are briefly considered; the simple designation system for nonrigid molecular entities is also outlined.

Algebraic Chirality Criteria and Their Application to Chirality Classification in Rigid Molecular Systems

Enumeration of cyclic graphs and cyclic designs

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Enumeration of cyclic graphs and cyclic designs