SUMMARY: The FASTQ+ format is designed for single-cell experiments. It extends various optional tags, including cell barcodes and unique molecular identifiers, to the sequence identifier and is fully compatible with the FASTQ format. In addition, PISA implements various utilities for processing sequences in the FASTQ format and alignments in the SAM/BAM/CRAM format from single-cell experiments, such as converting FASTQ format to FASTQ+, annotating alignments, PCR deduplication, feature counting and barcodes correction. The software is open-source and written in C language. AVAILABILITY AND IMPLEMENTATION: https://doi.org/10.5281/zenodo.7007056 or https://github.com/shiquan/PISA. SUPPLEMENTARY INFORMATION: Supplementary data are available at Bioinformatics online.

We study the eigenvalues of the connection Laplacian on a graph with an orthogonal group or unitary group signature. We establish higher order Buser type inequalities, i.e., we provide upper bounds for eigenvalues in terms of Cheeger constants in the case of nonnegative Ricci curvature. In this process, we discuss the concepts of Cheeger type constants and a discrete Ricci curvature for connection Laplacians and study their properties systematically. The Cheeger constants are defined as mixtures of the expansion rate of the underlying graph and the frustration index of the signature. The discrete curvature, which can be computed efficiently via solving semidefinite programming problems, has a characterization by the heat semigroup for functions combined with a heat semigroup for vector fields on the graph.

We study the Ollivier--Ricci curvature of graphs as a function of the chosen idleness. We show that this idleness function is concave and piecewise linear with at most three linear parts, and at most two linear parts in the case of a regular graph. We then apply our result to show that the idleness function of the Cartesian product of two regular graphs is completely determined by the idleness functions of the factors.

By assigning a probability measure via the spectrum of the normalized Laplacian to each graph and using Lp Wasserstein distances between probability measures, we define the corresponding spectral distances dp on the set of all graphs. This approach can even be extended to measuring the distances between infinite graphs. We prove that the diameter of the set of graphs, as a pseudo-metric space equipped with d1, is one. We further study the behavior of d1 when the size of graphs tends to infinity by interlacing inequalities aiming at exploring large real networks. A monotonic relation between d1 and the evolutionary distance of biological networks is observed in simulations.