Pseudograph and its associated real toric manifold

TL;DR

This paper extends the study of graph associahedra to pseudograph associahedra, providing methods to compute the Poincaré polynomial of the associated real toric varieties, thus broadening understanding of their topological properties.

Contribution

It introduces a way to compute the Poincaré polynomial for real toric varieties linked to pseudograph associahedra, expanding the scope beyond simple graphs.

Findings

Derived a method for computing the Poincaré polynomial of the associated real toric variety.

Extended existing results from simple graphs to pseudographs.

Enhanced understanding of the topological invariants of pseudograph associahedra.

Abstract

Given a simple graph $G$, the graph associahedron $P_G$ is a convex polytope whose facets correspond to the connected induced subgraphs of $G$. Graph associahedra have been studied widely and are found in a broad range of subjects. Recently, S. Choi and H. Park computed the rational Betti numbers of the real toric variety corresponding to a graph associahedron under the canonical Delzant realization. In this paper, we focus on a pseudograph associahedron which was introduced by Carr, Devadoss and Forcey, and then discuss how to compute the Poincar\'{e} polynomial of the real toric variety corresponding to a pseudograph associahedron under the canonical Delzant realization.