  The convex hull of the set of the incidence vectors of the matchings of a
graph G is the matching polytope of the graph, M(G). The graph whose vertices
and edges are the vertices and edges of M(G) is the skeleton of the matching
polytope of G, denoted G(M(G)). Since the number of vertices of G(M(G)) is
huge, the structural properties of these graphs have been studied in particular
classes. In this paper, for an arbitrary graph G, we obtain a formulae to
compute the degree of a vertex of G(M(G)) and prove that the minimum degree of
G(M(G)) is equal to the number of edges of G. Also, we identify the vertices of
the skeleton with the minimum degree and characterize regular skeletons of the
matching polytopes.
