From the beginning, graph theory has been aimed at applications, entering computer science by way of such applications as the famous shortest part problem and the reliability analysis of memory. More recent research activities have focused on using graphs to describe existing structures (network science) and to model abstractions for applications such as data classification and image or speech recognition, among others. This book, edited by Fu and Ma, falls in the former category.

The papers in this collection apply the methods elaborated in classical and algebraic graph theory to analyze patterns in various contexts. The first chapter, “Multilevel Analysis of Attributed Graphs for Explicit Graph Embedding in Vector Spaces,” by Luqman et al., serves as a general introduction to the whole topic of using graphs to represent application structures that are mapped in special vector spaces and then transformed or used for inferring some properties. The next chapter, “Feature Grouping and Selection Over an Undirected Graph,” by Yang et al., also covers the basics of the fundamental methodologies, in this case for high-dimensional regression. Later chapters deal with pattern recognition, mainly in image analysis. In “Median Graph Computation by Means of Graph Embedding into Vector Spaces,” Ferrer et al. elaborate on the use of a median graph, which is a graph relatively most similar to the set of selected graphs. In “Patch Alignment for Graph Embedding,” Luo et al. introduce the method of manifold learning-based dimensionality reduction as used for image pattern analysis. Unsupervised classification is dealt with in “Improving Classifications Through Graph Embeddings,” by Chatterjee et al.

Next, in “Learning with ℓ¹-Graph for High Dimensional Data Analysis,” Yang et al. focus on a specific type of graph construction that takes into account the context and neighborhood of the data, with examples of applications related to image patterns. In “Graph-Embedding Discriminant Analysis on Riemannian Manifolds for Visual Recognition,” Shirazi et al. address the problem of image pattern analysis with non-Euclidean geometry. In “A Flexible and Effective Linearization Method for Subspace Learning,” Nie et al. present some new results in semi-supervised learning. In “A Multi-graph Spectral Framework for Mining Multi-source Anomalies,” Gao et al. give some methods for anomaly detection, as the title suggests. The last chapter jumps to a somewhat different application area: Karam and Campbell present “Graph Embedding for Speaker Recognition.”

This book is certainly only for experts. While the depth of the mathematical material varies from chapter to chapter, a reader not well acquainted with graph theory will have to put a lot of effort into understanding most of the contents. On the other hand, the book will be easy for a researcher well versed in the theoretical fundamentals of the presented methods. As a collection of loosely connected works, it cannot be treated as a systematic lecture introducing the problem of graph embedding in a selected field. However, I do appreciate that, unlike many other books involving a set of works by various authors, for this volume, the editors have been able to structure the contents in an effective and interesting way. Therefore, I can recommend this volume as a useful reference for specialists in the field.