Extremal Betti Numbers and Persistence in Flag Complexes
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912252619128832 |
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| author | Beers, Lies Botnan, Magnus Bakke |
| author_facet | Beers, Lies Botnan, Magnus Bakke |
| contents | We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on $n$ vertices, we show that $β_k(X(G))$ is maximal when $G=\mathcal{T}_{n,k+1}$, the Turán graph on $k+1$ partition classes, where $X(G)$ denotes the flag complex of $G$. Building on this, we construct an edgewise (one edge at a time) filtration $\mathcal{G}=G_1\subseteq \cdots \subseteq \mathcal{T}_{n,k+1}$ for which $β_k(X(G_i))$ is maximal for all graphs on $n$ vertices and $i$ edges. Moreover, the persistence barcode $\mathcal{B}_k(X(G))$ achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with $|E(\mathcal{T}_{n,k+1})|$ edges.
For $k=1$, we consider edgewise filtrations of the complete graph $K_n$. We show that the maximal number of intervals in the persistence barcode is obtained precisely when $G_{\lceil n/2\rceil \cdot \lfloor n/2 \rfloor}=\mathcal{T}_{n,2}$. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize $β_1(X(G_i))$ for all $i$, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of $K_n$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_21294 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extremal Betti Numbers and Persistence in Flag Complexes Beers, Lies Botnan, Magnus Bakke Combinatorics Algebraic Topology Optimization and Control We investigate several problems concerning extremal Betti numbers and persistence in filtrations of flag complexes. For graphs on $n$ vertices, we show that $β_k(X(G))$ is maximal when $G=\mathcal{T}_{n,k+1}$, the Turán graph on $k+1$ partition classes, where $X(G)$ denotes the flag complex of $G$. Building on this, we construct an edgewise (one edge at a time) filtration $\mathcal{G}=G_1\subseteq \cdots \subseteq \mathcal{T}_{n,k+1}$ for which $β_k(X(G_i))$ is maximal for all graphs on $n$ vertices and $i$ edges. Moreover, the persistence barcode $\mathcal{B}_k(X(G))$ achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with $|E(\mathcal{T}_{n,k+1})|$ edges. For $k=1$, we consider edgewise filtrations of the complete graph $K_n$. We show that the maximal number of intervals in the persistence barcode is obtained precisely when $G_{\lceil n/2\rceil \cdot \lfloor n/2 \rfloor}=\mathcal{T}_{n,2}$. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize $β_1(X(G_i))$ for all $i$, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of $K_n$. |
| title | Extremal Betti Numbers and Persistence in Flag Complexes |
| topic | Combinatorics Algebraic Topology Optimization and Control |
| url | https://arxiv.org/abs/2502.21294 |