2012/02/22 by Guillaume Chapuy, Valentin Féray, Chapuy, Guillaume +1 · 3 citations
Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.1202.4976
openalex publication_date 2012/02/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Recently in graph theory several authors have studied the spectrum of the Cayley graph of the symmetric group Sn generated by the transpositions (1, i) for 2 <= i <= n. Several conjectures were made and partial results were obtained. The purpose of this note is to point out that, as mentioned also by P. Renteln, this problem is actually already solved in another context. Indeed it is equivalent to studying the spectrum of so-called Jucys-Murphy elements in the algebra of the symmetric group, which is well understood. The aforementioned conjectures are direct consequences of the existing theory. We also present a related result from P. Biane, giving an asymptotic description of this spectrum. We insist on the fact that this note does not contain any new results, but has only been written to convey the information from the algebraic combinatorics community to graph theorists.