Persistent homology is constrained to purely topological persistence, while multiscale graphs account only for geometric information. This work introduces persistent spectral theory to create a unified low-dimensional multiscale paradigm for revealing topological persistence and extracting geometric shapes from high-dimensional datasets. For a point-cloud dataset, a filtration procedure is used to generate a sequence of chain complexes and associated families of simplicial complexes and chains, from which we construct persistent combinatorial Laplacian matrices. We show that a full set of topological persistence can be completely recovered from the harmonic persistent spectra, that is, the spectra that have zero eigenvalues, of the persistent combinatorial Laplacian matrices. However, non-harmonic spectra of the Laplacian matrices induced by the filtration offer another powerful tool for data analysis, modeling, and prediction. In this work, fullerene stability is predicted by using both harmonic spectra and non-harmonic persistent spectra, while the latter spectra are successfully devised to analyze the structure of fullerenes and model protein flexibility, which cannot be straightforwardly extracted from the current persistent homology. The proposed method is found to provide excellent predictions of the protein B-factors for which current popular biophysical models break down.

Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze–Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze–Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each ε′>ε, an ε′-regular partition with k parts if there exists an ε-regular partition with k parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.

We introduce the notion of Bonnet-Myers and Lichnerowicz sharpness in the Ollivier Ricci curvature sense. Our main result is a classification of all self-centered Bonnet-Myers sharp graphs (hypercubes, cocktail party graphs, even-dimensional demi-cubes, Johnson graphs , the Gosset graph J(2n, n) and suitable Cartesian products). We also present a purely combinatorial reformulation of this result. We show that Bonnet-Myers sharpness implies Lichnerowicz sharpness and classify all distance-regular Lichnerowicz sharp graphs under the additional condition θ_1=b_1-1. We also relate Bonnet-Myers sharpness to an upper bound of Bakry-Émery ∞-curvature, which motivates a general conjecture about Bakry-Émery ∞-curvature.