Repr\'esentations lin\'eaires des graphes finis

TL;DR

This paper explores the relationship between isometric sheaves of linear lines in finite-dimensional vector spaces and associated graphs, describing their structures, symmetries, and examples.

Contribution

It characterizes all sheaves linked to a given graph and constructs their isometry groups as extensions of graph automorphism groups.

Findings

Complete classification of sheaves for a given graph

Construction of isometry groups as extension groups

Illustrative examples demonstrating the theory

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={U_i | i in X} of linear lines in E is an isometric sheaf, if there exist generators u_i of the lines U_i, and real constants ''omega'' and ''c '' such that : forall i,j in X, alpha(u_i,u_i)=omega, and if i is different from j, then alpha(u_i,u_j)=epsilon_{i,j}.c, with epsilon_i,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 epsilon_{i,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.