A decomposition of graph a-numbers
Suyuong Choi, Younghan Yoon
TL;DR
The paper develops a combinatorial decomposition for the $a$-sequence of a graph, tying graph invariants to the rational Betti numbers of the real toric variety $X^\mathbb{R}_G$ and providing a concrete edge-addition formula that expresses changes in $a_i(G)$ via reconnected complements. This yields monotonicity of $a_i$ under graph inclusion and sharp bounds for $a_i$ in terms of spanning trees and complete graphs, with explicit Cat-alan-type formulas for paths and Euler zigzag numbers for stars. It further proves unimodality of the $a$-sequence for broad classes (e.g., graphs with Hamiltonian circuits or a universal vertex), while acknowledging that log-concavity does not generally hold and that the unimodality phenomenon extends beyond classical toric manifolds. The work also discusses toric blow-ups, real loci, and the absence of Lefschetz-type operators in these real settings, illustrating rich interactions between combinatorics, topology, and real algebraic geometry.
Abstract
We study the $a$-sequence $(a_0(G), a_1(G), \cdots)$ of a finite simple graph $G$, defined recursively through a combinatorial rule and known to coincide with the sequence of rational Betti numbers of the real toric variety associated with $G$. In this paper, we establish a combinatorial and topological decomposition formula for the $a$-sequence. As an application, we show that the $a$-sequence is monotone under graph inclusion; that is, $a_i(G) \geq a_i(H)$ for all $i \geq 0$ whenever $H$ is a subgraph of $G$, and obtain the lower and upper bounds of $a_i$-numbers. We also prove that the $a$-sequence is unimodal in $i$ for a broad class of graphs $G$, including those with a Hamiltonian circuit or a universal vertex. These results provide a new class of topological spaces whose Betti number sequences are unimodal but not necessarily log concave, contributing to the study of real loci in algebraic geometry.