Spectral graph theory

About: Spectral graph theory is a research topic. Over the lifetime, 1334 publications have been published within this topic receiving 77373 citations.

Spectral Graph Optimization for Instance Reduction

Abstract: The operation of instance-based learning algorithms is based on storing a large set of prototypes in the system's database. However, such systems often experience issues with storage requirements, sensitivity to noise, and computational complexity, which result in high search and response times. In this brief, we introduce a novel framework that employs spectral graph theory to efficiently partition the dataset to border and internal instances. This is achieved by using a diverse set of border-discriminating features that capture the local friend and enemy profiles of the samples. The fused information from these features is then used via graph-cut modeling approach to generate the final dataset partitions of border and nonborder samples. The proposed method is referred to as the spectral instance reduction (SIR) algorithm. Experiments with a large number of datasets show that SIR performs competitively compared to many other reduction algorithms, in terms of both objectives of classification accuracy and data condensation.

The Cheeger constant of a quantum graph: The Cheeger constant of a quantum graph

Abstract: We review the theory of Cheeger constants for graphs and quantum graphs and their present and envisaged applications.

Dynamic Graph Algorithms and Graph Sparsification: New Techniques and Connections.

Abstract: Graphs naturally appear in several real-world contexts including social networks, the web network, and telecommunication networks. While the analysis and the understanding of graph structures have been a central area of study in algorithm design, the rapid increase of data sets over the last decades has posed new challenges for designing efficient algorithms that process large-scale graphs. These challenges arise from two usual assumptions in classical algorithm design, namely that graphs are static and that they fit into a single machine. However, in many application domains, graphs are subject to frequent changes over time, and their massive size makes them infeasible to be stored in the memory of a single machine. Driven by the need to devise new tools for overcoming such challenges, this thesis focuses on two areas of modern algorithm design that directly deal with processing massive graphs, namely dynamic graph algorithms and graph sparsification. We develop new algorithmic techniques from both dynamic and sparsification perspective for a multitude of graph-based optimization problems which lie at the core of Spectral Graph Theory, Graph Partitioning, and Metric Embeddings. Our algorithms are faster than any previous one and design smaller sparsifiers with better (approximation) quality. More importantly, this work introduces novel reduction techniques that show unexpected connections between seemingly different areas such as dynamic graph algorithms and graph sparsification.