This paper is concerned with convexity properties of m X n matrices whose entries are non-negative real numbers and whose row and column sums are specified positive numbers. Such matrices make their appearance in an irapor tan t linear programming problem known as the Hitchcock or t ransportat ion problem. The determination of the vertices of such sets of matrices enables one to obtain all solutions of the t ransportat ion problem when the solution is not uniquely determined. Let r~, r2, , r ~ , cl , c~, , c~ be positive numbers such tha t ~ m r, ~ 1 c~ . Call R = ( h , r2, . , rm) and C = (cl, c~, . . , c~) row and column sum vectors. Let ~I(R, C) be the class of all m × n matrices A with non-negative entries such tha t the sums of the entries in the i th row and j t h column of A are r, and ca respectively. The set of matrices ~I(R, C) form a convex set. A matr ix of A of the set ~I(R, C) is called a vertex matrix if there do not exist matrices B a n d C i n ~I(R,C) a n d a n u m b e r a, 0 0 f o r i = 1 , 2 , . , r , ~ : _ ~ M = 1 and A = ~ : = ~ X~Bi. Our main result is an algorithm for expressing a matr ix A of the set ~I(R, C) as an average of vertex matrices, together with an algorithm for the construction of vertex matrices. Some special results are exhibited in the case of generalized doubly stochastic matrices, i.e. matrices for which r~ = r2 = ' ' = rm and cl = c2 = • . . = an. There is an int imate connection between matrices with non-negative entries and bipart i te graphs. By a bipartite graph we mean the following structure. There is a system K consisting of three sets; two vertex sets S and T (whose elements are denoted by s~ and t~ respectively) and a set of edges E which is a subset of the Cartesian product S X T. Each edge of E will be denoted by a pair (s, t) with s in S and t in T. The term rank of a graph K is the largest number p of edges in a set (s~, t~) (s2, t2) (s3, t~) . . (sp, tp), no two of which have a vertex in common. Any such set of p edges is called a maximal set of independent edges. In connection with a bipart i te graph K, the concepts of minimal covering, reducibility, connectivity, and cycle play an impor tant role, and we define these as follows: Let A be a subset of S, and B a subset of T. The pair [A, B] is said to cover