Discrete polymatroids are the multi-set analogue of matroids. In this paper, we explore the connections among linear network coding, linear index coding and representable discrete polymatroids. We consider vector linear solutions of networks over a field @math with possibly different message and edge vector dimensions, which are referred to as linear fractional solutions. We define a network and show that a linear fractional solution over a field @math exists for a network if and only if the network is discrete polymatroidal with respect to a discrete polymatroid representable over @math An algorithm to construct networks starting from certain class of discrete polymatroids is provided. Every representation over @math for the discrete polymatroid, results in a linear fractional solution over @math for the constructed network. Next, we consider the index coding problem and show that a linear solution to an index coding problem exists if and only if there exists a representable discrete polymatroid satisfying certain conditions which are determined by the index coding problem considered. El Rouayheb et. al. showed that the problem of finding a multi-linear representation for a matroid can be reduced to finding a for an index coding problem obtained from that matroid. We generalize the result of El Rouayheb et. al. by showing that the problem of finding a representation for a discrete polymatroid can be reduced to finding a perfect linear index coding solution for an index coding problem obtained from that discrete polymatroid.
 Determining the achievable rate region for networks using routing, linear coding, or non-linear coding is thought to be a difficult task in general, and few are known. We describe the achievable rate regions for four interesting networks (completely for three and partially for the fourth). In addition to the known matrix-computation method for proving outer bounds for linear coding, we present a new method which yields actual characteristic-dependent linear rank inequalities from which the desired bounds follow immediately. We define the routing capacity of a network to be the supremum of all possible fractional message throughputs achievable by routing. We prove that the routing capacity of every network is achievable and rational, we present an algorithm for its computation, and we prove that every rational number in (0, 1] is the routing capacity of some solvable network. We also determine the routing capacity for various example networks. Finally, we discuss the extension of routing capacity to fractional coding solutions and show that the coding capacity of a network is independent of the alphabet used