Reflexive polytopes arising from bipartite graphs with $$\gamma $$-positivity associated to interior polynomials

Abstract

In this paper, we introduce polytopes $${\mathscr {B}}_G$$ B G arising from root systems $$B_n$$ B n and finite graphs G, and study their combinatorial and algebraic properties. In particular, it is shown that $${\mathscr {B}}_G$$ B G is reflexive if and only if G is bipartite. Moreover, in the case, $${\mathscr {B}}_G$$ B G has a regular unimodular triangulation. This implies that the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is palindromic and unimodal when G is bipartite. Furthermore, we discuss stronger properties, namely the $$\gamma $$ γ -positivity and the real-rootedness of the $$h^*$$ h ∗ -polynomials. In fact, if G is bipartite, then the $$h^*$$ h ∗ -polynomial of $${\mathscr {B}}_G$$ B G is $$\gamma $$ γ -positive and its $$\gamma $$ γ -polynomial is given by an interior polynomial (a version of the Tutte polynomial for a hypergraph). The $$h^*$$ h ∗ -polynomial is real-rooted if and only if the corresponding interior polynomial is real-rooted. From a counterexample to Neggers–Stanley conjecture, we construct a bipartite graph G whose $$h^*$$ h ∗ -polynomial is not real-rooted but $$\gamma $$ γ -positive, and coincides with the h-polynomial of a flag triangulation of a sphere.

Symmetric edge polytopes and matching generating polynomials

“…Remark 7.6. It follows from the proof of [26,Theorem 2.6] that for any forest G, ∂B G has a flag unimodular triangulation.…”

“…In [26], lattice polytopes arising from the root system of type B n and finite graphs were introduced. The symmetric edge polytope B G of type B of a graph G on the vertex set [n] is the lattice polytope which is the convex hull of…”

Symmetric decompositions, triangulations and real-rootedness

“…is a well studied polynomial of degree at most n with nonnegative coefficients, called the h * -polynomial of P. Stapledon [43] showed that h * (P, x) has a nonnegative symmetric decomposition with respect to n whenever P contains a lattice point in its relative interior (and in fact, to the best of our knowledge, the concept of a symmetric decomposition first appeared in [43]). While the question of unimodality of the h * -polynomial has long been studied [18], its γ-positivity and real-rootedness have been investigated more recently for several special classes of lattice polytopes [12,22,28,29,35,36,38] and the unimodality and real-rootedness of its symmetric decomposition have been addressed too [17,30,37].…”

Symmetric edge polytopes and matching generating polynomials

“…In [25], lattice polytopes arising from the root system of type B n and finite graphs were introduced. The symmetric edge polytope B G of type B of a graph G on the vertex set [n] is the lattice polytope which is the convex hull of…”