Repr\'esentations lin\'eaires des graphes finis
Abstract
Let X be a non-empty finite set and alpha a symmetric bilinear form on a real finite dimensional vector space E. We say that a set GG=Ui | i in X of linear lines in E is an isometric sheaf, if there exist generators ui of the lines Ui, and real constants ''omega'' and ''c '' such that : forall i,j in X, alpha(ui,ui)=omega, and if i is different from j, then alpha(ui,uj)=epsiloni,j.c, with epsiloni,j in -1,+1 Let Gamma be the graph whose set of vertices is X, two of them, say i and j, being linked when epsiloni,j = - 1. In this article we explore the relationship between GG and Gamma ; we describe all sheaves associated with a given graph Gamma and construct the group of isometries stabilizing one of those as an extension group of Aut(Gamma). We finally illustrate our construction with some examples.