Construction d'une famille finie de graphes de Ramanujan

Abstract

In this article, we intuitively define and introduce Ramanujan graphs. We focus on notable Ramanujan graphs known as sum graphs, characterized by specific properties related to eigenvalues. Indeed, studying these eigenvalues allows us to determine the spectrum of a Ramanujan graph, which is defined as the minimum of the difference between k and the absolute value (other than k) of an eigenvalue of a k-regular Ramanujan graph. This spectrum is as large as possible. For a graph to be Ramanujan, it is necessary and sufficient to study its eigenvalues, and eventually, to construct a finite family of Ramanujan graphs.