Related papers: Recent Colorings And Labelings In Topological Codi…
In this paper, we use a topological quantum field theory (TQFT) to define families of new homology theories of a $2$-dimensional CW complex of a smooth closed surface. The dimensions of these homology groups can be used to count the number…
Network Coding encourages information coding across a communication network. While the necessity, benefit and complexity of network coding are sensitive to the underlying graph structure of a network, existing theory on network coding often…
Our purpose is to show that complements of line graphs enjoy nice coloring properties. We show that for all graphs in this class the local and usual chromatic numbers are equal. We also prove a sufficient condition for the chromatic number…
Graphs are a basic tool for the representation of modern data. The richness of the topological information contained in a graph goes far beyond its mere interpretation as a one-dimensional simplicial complex. We show how topological…
In this paper, we introduce a generalization of graphlets to heterogeneous networks called typed graphlets. Informally, typed graphlets are small typed induced subgraphs. Typed graphlets generalize graphlets to rich heterogeneous networks…
Given their potential for fault-tolerant operations, topological quantum states are currently the focus of intense activity. Of particular interest are topological quantum error correction codes, such as the surface and planar stabilizer…
Graph coloring is fundamental to distributed computing. We give the first sub-logarithmic distributed algorithm for coloring cluster graphs. These graphs are obtained from the underlying communication network by contracting nodes and edges,…
Graph Neural Networks (GNNs) are learning models aimed at processing graphs and signals on graphs. The most popular and successful GNNs are based on message passing schemes. Such schemes inherently have limited expressive power when it…
More than ten years ago in 2008, a new kind of graph coloring appeared in graph theory, which is the {\it rainbow connection coloring} of graphs, and then followed by some other new concepts of graph colorings, such as {\it proper…
Recent advances in molecular representation learning have produced highly effective encodings of molecules for numerous cheminformatics and bioinformatics tasks. However, extracting general chemical insight while balancing predictive…
Color codes are topological stabilizer codes with unusual transversality properties. Here I show that their group of transversal gates is optimal and only depends on the spatial dimension, not the local geometry. I also introduce a…
This paper has dual aims. First is to develop practical universal coding methods for unlabeled graphs. Second is to use these for graph anomaly detection. The paper develops two coding methods for unlabeled graphs: one based on the degree…
Higher dimensional graphs can be used to colour two-dimensional geometric graphs. If G the boundary of a three dimensional graph H for example, we can refine the interior until it is colourable with 4 colours. The later goal is achieved if…
We demonstrate that graphs embedded on surfaces are a powerful and practical tool to generate, characterize and simulate networks with a broad range of properties. Remarkably, the study of topologically embedded graphs is non-restrictive…
Inferring topological and geometrical information from data can offer an alternative perspective on machine learning problems. Methods from topological data analysis, e.g., persistent homology, enable us to obtain such information,…
In this paper, we show that a simple coloring scheme can improve, both theoretically and empirically, the expressive power of Message Passing Neural Networks(MPNNs). More specifically, we introduce a graph neural network called Colored…
We study finite graphs embedded in oriented surfaces by associating a polynomial to it. The tools used in developing a theory of such graph polynomials are algebraic topological while the polynomial itself is inspired from ideas arising in…
Vertex coloring and multicoloring of graphs are a well known subject in graph theory, as well as their applications. In vertex multicoloring, each vertex is assigned some subset of a given set of colors. Here we propose a new kind of vertex…
In colored graphs, node classes are often associated with either their neighbors class or with information not incorporated in the graph associated with each node. We here propose that node classes are also associated with topological…
We propose a novel method for topological analysis of unweighted graphs which is based on \textit{persistent homology}. The proposed method maps the input graph to a complete weighted graph where the weighting function maps each edge to a…