Abstract
Heilmann et Lieb ont introduit le polynome de couplage µ(G, x) d'un graphe G =( V, E). Nous prolongeons leur definition en munissant chaque sommet de G d'une forme lineaire N -dimensionnelle (ou bien d'un vecteur) et chaque arete d'une forme symetrique bilineaire. On attache doncoutr-couplage de G le produit des formes lineaires des sommets qui ne sont pas satures par le couplage, multiplie par le produit des poids des r aretes du couplage, ou le poids d'une arete est la valeur de sa forme ´ ee sur les deux vecteurs de ses extremites. En multipliant par (−1) r et en sommant sur tous les couplages, nous obtenons notre polynome de couplage N -dimensionnel. Si N =1 , le theme principal de l'article de Heilmann et Lieb affirme que tous les zeros de µ(G, x )s ont reels. Si N =2 , cependant, nous avons trouve des graphes exceptionnels ou il n'y a aucun zero reel, meme si chaque arete est munie du produit scalaire canonique. Toutefois, la theorie de la dualitedee dans (La1) reste valable en N dimensions. Elle donne notamment une nouvelle interpretationa transformation de Bargmann-Segal, aux diagrammes de Feynman et aux produits de Wick. Heilmann and Lieb have introduced the matching polynomial µ(G, x) of a graph G =( V, E). We extend their definition by associating to every vertex of G an N -dimensional linear form (or a vector) and to every edge a symmetric bilinear form. For every r-matching of G we define its weight as the product of the linear forms of the vertices not covered by the matching, multiplied by the product of the weights of the r edges of the matching, where the weight of an edge is the value of its form evaluated at the two vectors of its end points. Multiplying by (−1) r and summing over all matchings, we get our N -dimensional matching polynomial. If N =1 , the Heilmann-Lieb theorem affirms that all zeroes of µ(G, x) are real. If N =2 , however, there are exceptional graphs without any real zero at all, even if the canonical scalar product is associated to every edge. Nevertheless, the duality theory developed in (La1) remains valid in N dimensions. In particular, it brings new light to the Bargmann-Segal transform, to the Feynman diagrams, and to the Wick products.
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