Repr\'esentations lin\'eaires des graphes finis

Publication date: 11 February 2009

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.

Mathematics Subject Classification ID

Finite automorphism groups of algebraic, geometric, or combinatorial structures (20B25) Graphs and abstract algebra (groups, rings, fields, etc.) (05C25) Graphs and linear algebra (matrices, eigenvalues, etc.) (05C50) Graph representations (geometric and intersection representations, etc.) (05C62) General theory for finite permutation groups (20B05) Multiply transitive finite groups (20B20)

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