A decomposition of graph a-numbers

TL;DR

This paper introduces a combinatorial and topological decomposition of the a-sequence of a graph, revealing monotonicity and unimodality properties, and connects these to Betti numbers of real toric varieties.

Contribution

It provides a new decomposition formula for the a-sequence, establishes its monotonicity under graph inclusion, and proves unimodality for broad classes of graphs.

Findings

a-sequence is monotone under subgraph inclusion

a-sequence is unimodal for graphs with Hamiltonian circuits or universal vertices

Betti number sequences can be unimodal without being log-concave

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.