Extremal Betti Numbers and Persistence in Flag Complexes

TL;DR

The article studies extremal questions for Betti numbers and persistence in filtrations of flag complexes, showing that Turán graphs $\mathcal{T}_{n,k+1}$ maximize the $k$-th flag Betti number and that there exist edgewise filtrations attaining these maxima at every step. It introduces fiberwise optimal filtrations that maximize both the Betti numbers and the total persistence, and provides a complete analysis of degree-1 persistence, including the maximal number of bars and the explicit optimal filtration structures. In the $k=1$ case, the maximal bar count is achieved precisely when the filtration ends at $\mathcal{T}_{n,2}$, with a near-complete classification of extremal filtrations, and the paper conjectures that these filtrations maximize total persistence over all edgewise filtrations of $K_n$. The results have implications for efficient persistent homology computations on flag complexes and offer a principled link between extremal graph theory (Turán graphs) and topological data analysis.

Abstract

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 $K_n$.

Figures (10)

• Figure 1: A graph (left), its flag complex (middle) and independence complex (right).
• Figure 2: An edgewise filtration of $K_5$.
• Figure 3: From left to right: the Turán graphs $\mathcal{T}_{5,2}$, $\mathcal{T}_{8,2}$ and $\mathcal{T}_{8,3}$.
• Figure 4: The filtration $\mathcal{H}$ of $\mathcal{T}_{8,2}$ from Definition \ref{['def:H']}.
• Figure 5: An illustration of Case $2$ from the proof of Theorem \ref{['thm:turanoptfilt']}.

Theorems & Definitions (69)

• Remark 1
• Example 2
• Lemma 3
• Lemma 4
• Proposition 5
• proof
• Definition 6
• Example 7
• Proposition 8
• Lemma 9