Extremal Betti Numbers and Persistence in Flag Complexes

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 = 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 𝒢 = G₁ ⊆ ⋯ ⊆ 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 ℬ_k(X(G)) achieves a maximal number of intervals, and total persistence, among all edgewise filtrations with |E(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_{⌈n/2⌉ ⋅ ⌊n/2⌋} = 𝒯_{n,2}. Among such filtrations, we characterize those achieving maximal total persistence. We further show that no filtration can optimize β₁(X(G_i)) for all i, and conjecture that our filtrations maximize the total persistence over all edgewise filtrations of K_n.