Low Earth Orbit satellite networks pose significant challenges to multi-hop semantic transmission because rapidly changing topology, link variability, and queue dynamics make end-to-end performance jointly depend on routing, relay processing, and semantic payload adaptation. Existing studies usually optimize routing or semantic transmission separately and are therefore not well suited to dynamic satellite scenarios under local observations. To address this issue, this paper proposes GraphJSCR, a graph-based joint routing and semantic coding method for multi-hop semantic transmission in dynamic Low Earth Orbit satellite networks. The satellite constellation is modeled as a time-varying directed graph, and the forwarding process is formulated as a partially observable sequential decision problem. A graph representation learning module is designed to encode local topology, link status, queue conditions, packet context, and semantic transmission states. Based on the learned representation, the proposed decision network jointly determines next-hop selection, relay processing level, and semantic transmission budget to balance end-to-end semantic quality and transmission delay. The semantic encoder-decoder is developed with reference to the SwinJSCC framework. Simulation results demonstrate that GraphJSCR achieves faster convergence and a better tradeoff between semantic fidelity and transmission efficiency than benchmark methods.

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