Explicit Construction of Ramanujan Bigraphs

Ballantine, Cristina; Feigon, Brooke; Ganapathy, Radhika; Kool, Janne; Maurischat, Kathrin; Wooding, Amy

doi:10.1007/978-3-319-17987-2_1

Cited by 12 publications

(14 citation statements)

References 21 publications

(25 reference statements)

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“…In addition, we have also shown how to construct the adjacency matrices for the currently available families of Ramanujan graphs. These explicit constructions, as well as forthcoming ones based on [5,14], are available for only a few combinations of degree and size. In contrast, it is known from [24,25] that Ramanujan graphs are known to exist for all degrees and all sizes.…”

Section: Discussionmentioning

confidence: 99%

“…The existence of Ramanujan bigraphs of specified degrees and sizes has been well studied in recent years. A "road map" for constructing Ramanujan bigraphs is given in [4], and some abstract constructions of Ramanujan bigraphs are given in [5,14]. These bigraphs have degrees ( p + 1, p 3 + 1) for various values of p, such as p ≡ 5 mod 12, p ≡ 11 mod 12 [5], and p ≡ 3 mod 4 [14].…”

Section: Two New Families Of Unbalanced Ramanujan Bigraphsmentioning

confidence: 99%

“…A "road map" for constructing Ramanujan bigraphs is given in [4], and some abstract constructions of Ramanujan bigraphs are given in [5,14]. These bigraphs have degrees ( p + 1, p 3 + 1) for various values of p, such as p ≡ 5 mod 12, p ≡ 11 mod 12 [5], and p ≡ 3 mod 4 [14]. 3 For each suitable choice of p, these papers lead to an infinite family of Ramanujan bigraphs.…”

Section: Two New Families Of Unbalanced Ramanujan Bigraphsmentioning

confidence: 99%

“…We state and prove two such explicit constructions, namely (lq, q 2 )-biregular graphs where q is any prime and l is any integer that satisfies 2 ≤ l ≤ q, and (q 2 , lq)-biregular graphs where q is any prime and l is any integer greater than q. Thus we can construct Ramanujan bigraphs for a broader range of degree-pairs as compared to [5,14]. In particular, for l = q we generate a new class of Ramanujan graphs.…”

Section: Two New Families Of Unbalanced Ramanujan Bigraphsmentioning

confidence: 99%

“…In contrast, however, no explicit methods for constructing unbalanced Ramanujan bigraphs are known. There are a couple of abstract constructions in [5,14], but these are not explicit. 1 This article has three goals.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

New and explicit constructions of unbalanced Ramanujan bipartite graphs

Burnwal¹,

Sinha

Vidyasagar³

2021

Ramanujan J

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The objectives of this article are threefold. Firstly, we present for the first time explicit constructions of an infinite family of unbalanced Ramanujan bigraphs. Secondly, we revisit some of the known methods for constructing Ramanujan graphs and discuss the computational work required in actually implementing the various construction methods. The third goal of this article is to address the following question: can we construct a bipartite Ramanujan graph with specified degrees, but with the restriction that the edge set of this graph must be distinct from a given set of “prohibited” edges? We provide an affirmative answer in many cases, as long as the set of prohibited edges is not too large.

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Section: Discussionmentioning

confidence: 99%

Section: Two New Families Of Unbalanced Ramanujan Bigraphsmentioning

confidence: 99%

Section: Two New Families Of Unbalanced Ramanujan Bigraphsmentioning

confidence: 99%

Section: Two New Families Of Unbalanced Ramanujan Bigraphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

New and explicit constructions of unbalanced Ramanujan bipartite graphs

Burnwal¹,

Sinha

Vidyasagar³

2021

Ramanujan J

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Ramanujan Complexes and Golden Gates in PU(3)

Evra

Parzanchevski

2022

Geom. Funct. Anal.

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The Clifford+T gate set is a topological generating set for P U (2), which has been wellstudied from the perspective of quantum computation on a single qubit. The discovery that it generates a full S-arithmetic subgroup of P U (2) has led to a fruitful interaction between quantum computation and number theory, leading in particular to a proof that words in these gates cover P U (2) in an almost-optimal manner.In this paper we study an analogue gate set for P U (3) called Clifford+D. We show that this set generates a full S-arithmetic subgroup of P U (3), and satisfies a slightly weaker almost-optimal covering property. Our proofs are different from those for P U (2): while both gate sets act naturally on a (Bruhat-Tits) tree, in P U (2) the generated group acts transitively on the vertices of the tree, and this is a main ingredient in proving both arithmeticity and efficiency. In the P U (3) (Clifford+D) case the action on the tree is far from being transitive. This makes the proof of arithmeticity considerably harder, and the study of covering rate by automorphic representation theory becomes more involved and results in a slower covering rate.

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From Ramanujan graphs to Ramanujan complexes

Lubotzky

Parzanchevski

2019

Phil. Trans. R. Soc. A.

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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 are surveyed. After explaining their connection to the Ramanujan conjecture, we will present some old and new results with an emphasis on random walks on these discrete objects and on the Euclidean spheres. The latter lead to ‘golden gates’ which are of importance in quantum computation. This article is part of a discussion meeting issue ‘Srinivasa Ramanujan: in celebration of the centenary of his election as FRS’.

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Explicit Construction of Ramanujan Bigraphs

Cited by 12 publications

References 21 publications

New and explicit constructions of unbalanced Ramanujan bipartite graphs

New and explicit constructions of unbalanced Ramanujan bipartite graphs

Ramanujan Complexes and Golden Gates in PU(3)

From Ramanujan graphs to Ramanujan complexes

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