This chapter discusses the elementary properties of Laplace operators on graphs and hypergraphs. Many interesting examples will illustrate how special eigenvalues emerge. We also introduce discrete Pólya–Cheeger constants and their dual versions and provide the initial steps relating spectral clustering, spectra of neighborhood graphs, signed Laplacians, and spectra of simplicial complexes and hypergraphs.

The strengths of this book include the directness of its encounter with research data, its advice on practical data analysis issues, careful critiques of analysis results, its use of modern data analysis tools and approaches, its use of simulation and other computer-intensive methods where these provide insight or give results that are not otherwise available, its attention to graphical and other presentation issues, its use of examples drawn from across the range of statistical applications, the links that it makes into the debate over reproducibility in science, and the inclusion of code that reproduces analyses. The methods that we cover have wide application. The datasets, many of which have featured in published papers, are drawn from many different fields. They reflect a journey in learning and understanding, alike for the authors and for those with whom they have worked, that has ranged widely over many different research areas. The R system has brought into a common framework a huge range of abilities for data analysis, data manipulation and graphics. Our text has as its aim helping its readers to take full advantage of those abilities.

We give a discussion of some basic mathematical concepts and terminologies connected with sets, numbers, functions and graphs.

Human history has created a large variety of sign systems for communication. These systems were developed at different times for different purposes. While oral language has developed as part of human biological evolution, written texts, realistic pictures, maps, and graphs are cultural inventions. Human oral language might have originated from gestures supplemented by sound patterns. It is a biological anchored feature of the human species, as manifested in somatic, perceptual, and neurological pre-adaptations. Early writing systems used iconic ideograms which were gradually transformed into symbols. This made production and discrimination easier but increased the required amount of learning. Further development led to writing systems using phonograms plus orthographic ideograms. Realistic pictures are older than writing systems. They represent content by similarity but also show allegories of social relationships. Maps are realistic pictures of a geographic area facing the problem of how to present a curved earth surface on a two-dimensional surface. Graphs are visuo-spatial objects representing a subject matter based on analogy due to inherent common structural properties.

The chapter on visual models discusses basic ways that scientists create visual representations of their data, including charts and graphs, in order to understand their data better. Like all models, visual models are a simplified version of reality. Two of the visual models discussed in this chapter are the frequency table and histogram. The histogram, in particular, is useful in the shape of the distribution of data, skewness, kurtosis, and the number of peaks. Other visual models in the social sciences include frequency polygons, bar graphs, stem-and-leaf plots, line graphs, pie charts, and scatterplots. All of these visual models help researchers understand their data in different ways, though none is perfect for all situations. Modern technology has resulted in the creation of new ways to visualize data. These methods are more complex, but they provide data analysts with new insights into their data. The incorporation of geographic data, animations, and interactive tools give people more options than ever existed in previous eras.

Chapter 6 covers global flat foldability.This includes determining how we can tell if a crease pattern with multiple vertices will fold flat without forcing the paper to self-intersect, as well as discovering properties that all such crease patterns have, beyond what was covered in the previous chapter.Justin’s Theorem, which is a generalization of Kawasaki’s and Maekawa’s Theorems and whose proof uses elements of basic knot theory, is covered, as are Justin’s non-crossing conditions that provide necessary and sufficient conditions for a general crease pattern to fold flat.The matrix model from Chapter 5 is developed further to create a formal folding map for flat origami.Finally, Bern and Hayes’ seminal proof that determining flat foldability of a given crease pattern is NP-Hard is presented and updated with more recent results on box pleating.

The field of flat-foldable origami is introduced, which involves a mix of geometry and combinantorics.This chapter focuses on local properties of flat origami, meaning the study of how and when a single vertex in an origami crease pattern will be able to fold flat.The classic theorems of Kawasaki and Maekawa are proved and generalizations are made to folding vertices on cone-shaped (i.e., non-developable) paper.The problem of counting valid mountain-valley assignments of flat-foldable vertices is solved, and the configuration space of flat-foldable vertices of a fixed degree is characterized.A matrix model for formalizing flat-vertex folds is introduced, and the chapter ends with historical notes on this topic.

In this expository article we give an overview of recent developments in the study of optimal Hardy-type inequalityin the continuum and in the discrete setting. In particular, we present the technique of the {\emph supersolution construction} that yield “as large as possibleȍ Hardy weightswhich is made precise in terms ofthe notion of criticality. Instead of presenting the most general setting possible, we restrict ourselves to the case of the Laplacian on smooth manifolds and bounded combinatorial graphs. Although the results hold in far greater generality, the fundamental phenomena as well as the core ideas of the proofs become especially clear in these basic settings.