Unimodular modules Original Research Article

The present publication is mainly a survey paper on the authorʹs contributions on the relations between graph theory and linear algebra. A system of axioms introduced by Ghouila-Houri allows one to generalize to an arbitrary Abelian group the notion of interval in a linearly ordered group and to state a theorem that generalizes two due to A.J. Hoffman. The first is on the feasibility of a system of inequalities and the other is Hoffmanʹs circulation theorem reported in the first Bergeʹs book on graph theory. Our point of view permitted us to prove classical results of linear programming in the more general setting of linearly ordered groups and rings. It also shed a new light on convex programming.