A new graph invariant arises in toric topology

TL;DR

This paper introduces new combinatorial invariants derived from graphs in toric topology, enabling the calculation of Betti numbers and Euler characteristics of associated real toric varieties through purely combinatorial methods.

Contribution

It defines novel graph invariants linked to toric topology and provides combinatorial formulas for topological invariants of graph associahedron-based varieties.

Findings

Invariants relate to Catalan and Euler zigzag numbers for specific graph families.

Betti numbers and Euler characteristic can be computed combinatorially.

New invariants reveal deep connections between graph theory and topology.

Abstract

In this paper, we introduce new combinatorial invariants of any finite simple graph, which arise in toric topology. We compute the $i$-th (rational) Betti number and Euler characteristic of the real toric variety associated to a graph associahedron $P_{\B(G)}$. They can be calculated by a purely combinatorial method (in terms of graphs) and are named $a_i(G)$ and $b(G)$, respectively. To our surprise, for specific families of the graph $G$, our invariants are deeply related to well-known combinatorial sequences such as the Catalan numbers and Euler zigzag numbers.