Combinatorial Theory

Symmetric edge polytopes \(\mathcal{A}_G\) of type A are lattice polytopes arising from the root system \(A_n\) and finite simple graphs \(G\). There is a connection between \(\mathcal{A}_G\) and the Kuramoto synchronization model in physics. In particular, the normalized volume of \(\mathcal{A}_G\) 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 \(\mathcal{A}_{\widehat{G}}\) by using matching generating polynomials, where \(\widehat{G}\) is the suspension of \(G\). This gives also a formula for the normalized volume of \(\mathcal{A}_{\widehat{G}}\). Moreover, via methods from chemical graph theory, we show that for any cactus graph \(G\), the \(h^*\)-polynomial of \(\mathcal{A}_{\widehat{G}}\) 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.

Keywords: Symmetric edge polytope, \(h^*\)-polynomial, interior polynomial, matching generating polynomial, \(\mu\)-polynomial, real-rooted, \(\gamma\)-positive.

Mathematics Subject Classifications: 05A15, 05C31, 13P10, 52B12, 52B20