Arithmetic aspects of symmetric edge polytopes

Abstract

We investigate arithmetic, geometric and combinatorial properties of symmetric edge polytopes. We give a complete combinatorial description of their facets. By combining Gröbner basis techniques, half-open decompositions and methods for interlacing polynomials we provide an explicit formula for the $h^{\ast }$-polynomial in case of complete bipartite graphs. In particular, we show that the $h^{\ast }$-polynomial is $\gamma$-positive and real-rooted. This proves Gal's conjecture for arbitrary flag unimodular triangulations in this case, and, beyond that, we prove a strengthing due to Nevo and Petersen (2011).

18 pages, 1 figure. Comments are welcome

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• The $h^{\ast }$-polynomials of locally anti-blocking lattice polytopes and their $\gamma$-positivity
• A toric deformation method for solving Kuramoto equations
• Reflexive polytopes arising from bipartite graphs with $\gamma$-positivity associated to interior polynomials
• On the gamma-vector of symmetric edge polytopes
• Root polytopes and Jaeger-type dissections for directed graphs
• PQ-type adjacency polytopes of join graphs
• Facets of Random Symmetric Edge Polytopes, Degree Sequences, and Clustering
• A geometric proof for the root-independence of the greedoid polynomial of Eulerian branching greedoids
• $h^{\ast }$-vectors of graph polytopes using activities of dissecting spanning trees
• Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs
• Gorenstein graphic matroids
• Rigid Gorenstein toric Fano varieties arising from directed graphs