Symmetric edge polytopes and matching generating polynomials

Abstract

Symmetric edge polytopes A G of type A are lattice polytopes arising from the root system A n and finite simple graphs G. There is a connection between A G and the Kuramoto synchronization model in physics. In particular, the normalized volume of A G plays a central role. In the present paper, we focus on a particular class of graphs. In fact, for any cactus graph G, we give a formula for the h * -polynomial of A G by using matching generating polynomials, where G is the suspension of G. This gives also a formula for the normalized volume of A G . Moreover, via the chemical graph theory, we show that for any cactus graph G, the h * -polynomial of A G is real-rooted. Finally, we extend the discussion to symmetric edge polytopes of type B, which are lattice polytopes arising from the root system B n and finite simple graphs.

PQ-type adjacency polytopes of join graphs

“…where g(C n , x) is the matching generating polynomial of C n . Moreover, for n ≥ 3, it is known [16,Example 4.5] that, γ(n, x) is the γ-polynomial of the PV-type adjacency polytope ∇ PV W n of W n . (Note that ∇ PV W n is called the symmetric edge polytope of type A of W n in [16].)…”

“…Moreover, for n ≥ 3, it is known [16,Example 4.5] that, γ(n, x) is the γ-polynomial of the PV-type adjacency polytope ∇ PV W n of W n . (Note that ∇ PV W n is called the symmetric edge polytope of type A of W n in [16].) The h * -polynomial of ∇ PQ W n is described by this function as follows.…”

Normalized Volumes of Type-PQ Adjacency Polytopes for Certain Classes of Graphs

“…The first author, together with Chen and others, have recently studied their combinatorial structures and triangulations with a view towards their use in homotopy continuation methods [3,4,7]. Their work has sparked a flurry of results [1,5,6,10,11] using various algebraic and combinatorial techniques, often under the name of symmetric edge polytopes, which were studied previously with a view towards Ehrhart-theoretic results [8,9].…”

Rigid Gorenstein toric Fano varieties arising from directed graphs

“…Notions of directed edge polytopes and symmetric edge polytopes coincide when G is a symmetric directed graph, i.e., A G = P G . Symmetric edge polytopes are of interest in many fields such as commutative algebra ( [18]), algebraic geometry ( [11]) algebraic combinatorics ( [12,19,20]) and number theory ( [6,15,22]). One of the reasons the symmetric edge polytopes gained much attraction is due to its connection to the Kuramoto model ( [14]), which describes the behavior of interacting oscillators (see [7]).…”