Some presentations of matroid and oriented matroid theories are available IN FRENCH:
here. In particular, there is a book chapter [L].
Note. Since matroids and oriented matroids generalize space cycles of graphs, results in these theories applie in particular to graphs.
This polynomial, coming from graph theory (1950') and extending the chromatic polynomial to two dual variables, generalizes very well to matroids and contains a lot of structural and enumerative information.
This object is underlying in several of my research:
[J4] (polynomial complexity of its evaluation at a point left partially open in Jaeger's paper), [J5][J6] (link with edge reversing dynamical systems) and overall those below.
The (attr)active mapping of ordered oriented matroids.
With Michel Las Vergnas,
we have defined
and studied a general and natural mapping which associates an ordered oriented matroid with one of its bases,
and which induces an activity (state models of the Tutte polynomial) preserving correspondence between bases and reorientations of a given ordered oriented matroid.
Several bijections, called active bijections appear as refinements or particular cases in several contexts.
It uses on the one hand duality
and decompositions of activities, and extends on the other hand the combinatorial formulation of linear programming.
For a more precise abstract taken from my thesis, click here .
See [C4] for a recent sum up of the whole construction in 7 pages,
[C8] for a recent sum up of the linear programming construction,
[C2] for an older but more complete survey (except linear programming).
For the detailed constructions, see
and also [J1] in the uniform case,
[J2][R1] in the graphical case,
et [J3] in the supersolvable case.
These constructions have been described or sketched in french in my thesis [T].