On Ramanujan bigraphs and general Ramanujan graphs

Shai Evra; Brooke Feigon; Kathrin Maurischat; Ori Parzanchevski

doi:10.5802/crmath.819

Article de synthèse - Combinatoire, Théorie des nombres

On Ramanujan bigraphs and general Ramanujan graphs
[Sur les bigraphes de Ramanujan et les graphes généraux de Ramanujan]

Shai Evra ¹ ; Brooke Feigon ² ; Kathrin Maurischat ³ ; Ori Parzanchevski ¹

¹Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
²Department of Mathematics, The City College of New York, CUNY, New York, NY 10031, USA
³Lehrstuhl für Algebra und Darstellungstheorie, RWTH Aachen University, 52062 Aachen, Germany

Comptes Rendus. Mathématique, Volume 364 (2026), pp. 71-78

Abstract (VO)
Résumé

In this paper, we give an overview of new results that define and explicitly construct Ramanujan Cayley biregular bipartite graphs. We also study the extremal combinatorial properties of these graphs. This parallels the work of Lubotzky, Phillips and Sarnak on regular Ramanujan Cayley graphs, with several interesting differences. Furthermore, this work also proposes a stronger definition of Ramanujan graphs than has been used in the past, which opens the door to future studies.

Dans cet article, nous donnons un aperçu des nouveaux résultats qui définissent et construisent explicitement les graphes bipartis biréguliers de Ramanujan Cayley. Nous étudions également les propriétés combinatoires extrêmes de ces graphes. Ces travaux sont parallèles à ceux de Lubotzky, Phillips et Sarnak sur les graphes de Ramanujan Cayley réguliers, avec plusieurs différences intéressantes. En outre, ce travail propose une définition plus forte des graphes de Ramanujan que celle utilisée dans le passé, ce qui ouvre la voie à de futures études.

Reçu le : 2025-03-21
Révisé le : 2025-11-25
Accepté le : 2026-01-22
Publié le : 2026-03-09

DOI : 10.5802/crmath.819

Classification : 11F70, 05C48
Keywords: Ramanujan graphs, Cayley bigraphs, non-backtracking spectrum, pseudorandomness, simply-transitive lattices, Ramanujan conjecture, automorphic representations of $\operatorname{U}(3)$
Mots-clés : Graphes de Ramanujan, bigraphes de Cayley, spectre non rétroactif, pseudo-aléas, treillis simplement transitifs, conjecture de Ramanujan, représentations automorphes de $\operatorname{U}(3)$

Affiliations des auteurs :

Shai Evra ¹ ; Brooke Feigon ² ; Kathrin Maurischat ³ ; Ori Parzanchevski ¹

¹ Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Israel
² Department of Mathematics, The City College of New York, CUNY, New York, NY 10031, USA
³ Lehrstuhl für Algebra und Darstellungstheorie, RWTH Aachen University, 52062 Aachen, Germany

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Shai Evra and Brooke Feigon and Kathrin Maurischat and Ori Parzanchevski},
     title = {On {Ramanujan} bigraphs and general {Ramanujan} graphs},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {71--78},
     year = {2026},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {364},
     doi = {10.5802/crmath.819},
     language = {en},
}

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Shai Evra; Brooke Feigon; Kathrin Maurischat; Ori Parzanchevski. On Ramanujan bigraphs and general Ramanujan graphs. Comptes Rendus. Mathématique, Volume 364 (2026), pp. 71-78. doi: 10.5802/crmath.819

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