Related papers: Ramanujan Graphs
Random graphs have proven to be one of the most important and fruitful concepts in modern Combinatorics and Theoretical Computer Science. Besides being a fascinating study subject for their own sake, they serve as essential instruments in…
This paper considers a higher-dimensional generalization of the notion of Ramanujan graphs, defined by Lubotzky, Phillips, and Sarnak. Specifically the Ramanujan property is studied for cubical complexes which are uniformized by an ordered…
We give an overview of different approaches to measuring the similarity of, or the distance between, two graphs, highlighting connections between these approaches. We also discuss the complexity of computing the distances.
This article is intended as a reference guide to various notions of monoidal categories and their associated string diagrams. It is hoped that this will be useful not just to mathematicians, but also to physicists, computer scientists, and…
Connected networks are a fundamental structure of neurobiology. Understanding these networks will help us elucidate the neural mechanisms of computation. Mathematically speaking these networks are `graphs' - structures containing objects…
This paper discusses some aspects of the history of the Paley graphs and their automorphism groups.
Ramanujan Master Theorem is a technique developed by the indian mathematician S. Ramanujan to evaluate a class of definite integrals. This technique is used here to calculate the values of integrals associated with specific Feynman…
In this dissertation, we explore the structure of inversion graphs of permutations--a class of graphs that naturally arises by representing each permutation as a graph, where vertices correspond to entries and edges encode inversions.…
Chain graphs combine directed and undirected graphs and their underlying mathematics combines properties of the two. This paper gives a simplified definition of chain graphs based on a hierarchical combination of Bayesian (directed) and…
We introduce a new notation for representing labeled regular bipartite graphs of arbitrary degree. Several enumeration problems for labeled and unlabeled regular bipartite graphs have been introduced. A general algorithm for enumerating all…
Gcd-graphs represent an interesting and historically important class of integral graphs. Since the pioneering work of Klotz and Sander, numerous incarnations of these graphs have been explored in the literature. In this article, we define…
We describe various path homology theories constructed for a directed hypergraph. We introduce the category of directed hypergraphs and the notion of a homotopy in this category. Also, we investigate the functoriality and the homotopy…
We are interested in overgroups of the automorphism group of the Rado graph. One class of such overgroups is completely understood; this is the class of reducts. In this article we tie recent work on various other natural overgroups, in…
$\omega$-periodic graphs are introduced and studied. These are graphs which arise as the limits of periodic extensions of the nearest neighbor graph on the integers. We observe that all bounded degree $\omega$-periodic graphs are ameanable.…
k-graphs are higher-rank analogues of directed graphs which were first developed to provide combinatorial models for operator algebras of Cuntz-Krieger type. Here we develop the theory of covering spaces for k-graphs, obtaining a…
Based on the previously proposed concept Understanding Tree, this paper introduces two concepts: Understanding Graph and Understanding Map, and explores their potential applications. Understanding Graph and Understanding Map can be deemed…
We give a brief review of the work of Ramanujan on cranks that is found in the Lost Notebook. Recent work by Bruce Berndt and his coauthors have brought to light many interesting results of Ramanujan on cranks, which we highlight in this…
Graph neural networks (GNNs) have demonstrated a significant boost in prediction performance on graph data. At the same time, the predictions made by these models are often hard to interpret. In that regard, many efforts have been made to…
In a famous paper of $1914$ Ramanujan gave a list of $17$ extraordinary formulas for the number $\pi$. In this paper we explain a general method to prove them, based on an original idea of James Wan and in some own ideas.
In this paper, we describe a reproduction of the Relational Graph Convolutional Network (RGCN). Using our reproduction, we explain the intuition behind the model. Our reproduction results empirically validate the correctness of our…