We study routing on a ring network in which traffic originates from nodes on the ring and is destined to the center. The users can take direct paths from originating nodes to the center and also multihop paths via other nodes. We show that routing games with only one and two hop paths and linear costs are potential games. We give explicit expressions of Nash equilibrium flows for networks with any generic cost function and symmetric loads. We also consider a ring network with random number of users at nodes, all of them having same demand, and linear routing costs. We give explicit characterization of Nash equilibria for two cases: (i) General i.i.d. loads and one and two hop paths, (ii) Bernoulli distributed loads. We also analyze optimal routing in each of these cases.