Related papers: Ramanujan Complexes and High Dimensional Expanders
Expander graphs have been a focus of attention in computer science in the last four decades. In recent years a high dimensional theory of expanders is emerging. There are several possible generalizations of the theory of expansion to…
Ramanujan graphs are graphs whose spectrum is bounded optimally. Such graphs have found numerous applications in combinatorics and computer science. In recent years, a high dimensional theory has emerged. In this paper these developments…
Expander graphs have been, during the last five decades, the subject of a most fruitful interaction between pure mathematics and computer science, with influence and applications going both ways (cf. [Lub94], [HLW06], [Lub12] and the…
Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to…
We present a generalization of the construction of graphs by Lubotzky, Phillips and Sarnak in their celebrated article "Ramanujan graphs". The new approach consists in using octonion algebras rather than quaternions. A key tool is the…
High dimensional expanders is a vibrant emerging field of study. Nevertheless, the only known construction of bounded degree high dimensional expanders is based on Ramanujan complexes, whereas one dimensional bounded degree expanders are…
Recently, a construction of minimal codes arising from a family of almost Ramanujan graphs was shown. Ramanujan graphs are examples of expander graphs that minimize the second-largest eigenvalue of their adjacency matrix. We call such…
This survey accompanies a lecture on the paper ``Interlacing Families I: Bipartite Ramanujan Graphs of All Degrees'' by A. Marcus, D. Spielman, and N. Srivastava at the 2024 International Congress of Basic Science (ICBS) in July, 2024. Its…
We discuss two combinatorical ways of generalizing the definition of expander graphs and Ramanujan graphs, to quotients of buildings of higher dimension. The two possible definitions are equivalent for affine buildings, giving the notion of…
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 define and construct Ramanujan complexes. These are simplicial complexes which are higher dimensional analogues of Ramanujan graphs. They are obtained as quotients of the buildings of type $\tilde{A}_{d-1}$ associated with…
Expander graphs have many interesting applications in communication networks and other areas, and thus these graphs have been extensively studied in theoretic computer sciences and in applied mathematics. In this paper, we use reversible…
Ramanujan complexes are high dimensional simplical complexes generalizing Ramanujan graphs. A result of Oh on quantitative property (T) for Lie groups over local fields is used to deduce a Mixing Lemma for such complexes. As an application…
The study of Ramanujan-type congruences for functions specific to additive number theory has a long and rich history. Motivated by recent connections between divisor sums and overpartitions via congruences in arithmetic progressions, we…
We generalize certain recent results of Ushiroya concerning Ramanujan expansions of arithmetic functions of two variables. We also show that some properties on expansions of arithmetic functions of one and several variables using classical…
Topics concerning metric dimension related invariants in graphs are nowadays intensively studied. This compendium of combinatorial and computational results on this topic is an attempt of surveying those contributions that are of the…
Expander graphs are widely used in communication problems and construction of error correcting codes. In such graphs, information gets through very quickly. Typically, it is not true for social or biological networks, though we may find a…
We study the Erdos distance problem over finite Euclidean and non-Euclidean spaces. Our main tools are graphs associated to finite Euclidean and non-Euclidean spaces that are considered in Bannai-Shimabukuro-Tanaka (2004, 2007). These…
This is an item on Ramanujan Graphs for a planned encyclopedia on Ramanujan. The notion of Ramanujan graphs is explained, as well as the reason to name these graphs after Ramanujan.
The question of finding expander graphs with strong vertex expansion properties such as unique neighbor expansion and lossless expansion is central to computer science. A barrier to constructing these is that strong notions of expansion…