A decomposition of graph a-numbers
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908828320137216 |
|---|---|
| author | Choi, Suyuong Yoon, Younghan |
| author_facet | Choi, Suyuong Yoon, Younghan |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_06855 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A decomposition of graph a-numbers Choi, Suyuong Yoon, Younghan Combinatorics Algebraic Geometry Algebraic Topology 14M25, 55N10, 14P25, 57S12, 05C30 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. |
| title | A decomposition of graph a-numbers |
| topic | Combinatorics Algebraic Geometry Algebraic Topology 14M25, 55N10, 14P25, 57S12, 05C30 |
| url | https://arxiv.org/abs/2508.06855 |