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In the present paper we study the power allocation for multi-hop underlay cognitive radio networks (CRNs) with line topology. That is, we consider a scenario, where a CRN operates in parallel to the primary network provided that the interference created to the primary network is limited by an acceptable threshold. The CNR is assumed to be a multi-hop relay network and hence before reaching the destination information from the source node may pass several hops from node to node. At each hop, the information is decoded and forwarded to the following node. It is further assumed that transmissions can be overheard by neighboring nodes, thereby creating interference. The power at each node can be optimally adjusted so that the end-to-end throughput of the CRN is maximized, while the constraint on interference towards the primary network is satisfied. In this paper, we show that for line CRNs the optimal power allocation is achieved when capacities of all intermediate links are equal and the interference constraint of the most disturbed primary user is fulfilled with equality. To simplify the computation of the optimal power allocation we derive two approximate solutions as well as several distributed power allocation schemes. The numerical results illustrate the interplay between the proposed solutions and the optimal power allocation.