Related papers: Expander Graphs in Pure and Applied Mathematics
Network-based modeling of complex systems and data using the language of graphs has become an essential topic across a range of different disciplines. Arguably, this graph-based perspective derives its success from the relative simplicity…
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for…
Processing large complex networks recently attracted considerable interest. Complex graphs are useful in a wide range of applications from technological networks to biological systems like the human brain. Sometimes these networks are…
Geospatial sciences include a wide range of applications, from environmental monitoring transportation to infrastructure planning, as well as location-based analysis and services. Graph theory algorithms in mathematics have emerged as…
Graphs are crucial for representing interrelated data and aiding predictive modeling by capturing complex relationships. Achieving high-quality graph representation is important for identifying linked patterns, leading to improvements in…
In this paper, we discuss how to design the graph topology to reduce the communication complexity of certain algorithms for decentralized optimization. Our goal is to minimize the total communication needed to achieve a prescribed accuracy.…
We construct explicit generating sets S_n and \tilde S_n of the for the alternating and the symmetric groups, which turn the Cayley graphs C(Alt(n), S_n) and C(Sym(n), \tilde S_n) into a family of bounded degree expanders for all n. This…
We introduce and study embeddings of graphs in finite projective planes, and present related results for some families of graphs including complete graphs and complete bipartite graphs. We also make connections between embeddings of graphs…
Edge expansion is a parameter indicating how well-connected a graph is. It is useful for designing robust networks, analysing random walks or information flow through a network and is an important notion in theoretical computer science.…
Particle physics is a branch of science aiming at discovering the fundamental laws of matter and forces. Graph neural networks are trainable functions which operate on graphs---sets of elements and their pairwise relations---and are a…
This book collects the lectures about graph theory and its applications which were given to students of mathematical departments of Moscow State University and Peking University. Graph theory is a very wide field with a lot of applications…
Graph Neural Networks (GNNs) have exploded onto the machine learning scene in recent years owing to their capability to model and learn from graph-structured data. Such an ability has strong implications in a wide variety of fields whose…
Random graphs are more and more used for modeling real world networks such as evolutionary networks of proteins. For this purpose we look at two different models and analyze how properties like connectedness and degree distributions are…
Dynamic graph learning has gained significant attention as it offers a powerful means to model intricate interactions among entities across various real-world and scientific domains. Notably, graphs serve as effective representations for…
Highly connected and yet sparse graphs (such as expanders or graphs of high treewidth) are fundamental, widely applicable and extensively studied combinatorial objects. We initiate the study of such highly connected graphs that are, in…
A matrix-weighted graph is an undirected graph with a $k\times k$ positive semidefinite matrix assigned to each edge. There are natural generalizations of the Laplacian and adjacency matrices for such graphs. These matrices can be used to…
We survey various aspects of infinite extremal graph theory and prove several new results. The lead role play the parameters connectivity and degree. This includes the end degree. Many open problems are suggested.
Graphs, and sequences of growing graphs, can be used to specify the architecture of mathematical models in many fields including machine learning and computational science. Here we define structured graph "lineages" (ordered by level…
Sparse graphs and their associated matroids play an important role in rigidity theory, where they capture the combinatorics of generically rigid structures. We define a new family called {\bf graded sparse graphs}, arising from generically…