Large language model (LLM)-based agents are increasingly becoming self-evolving systems that persist across interactions, maintain memories, use tools, acquire skills, refine workflows, and coordinate with other agents. These capabilities make agent states structural and dynamic: entities, relations, attributes, dependencies, and execution structures change with new evidence, feedback, and environmental conditions. Existing graph-agent surveys typically treat graphs as support structures for agent functions rather than as evolving substrates, while self-evolving-agent surveys focus on agent-level mechanisms and rarely discuss graph topology evolution. Thus, the coupling between evolving agent state and dynamic graph topology remains underexplored. This survey connects these two research lines by framing agent evolution as dynamic graph transformation. We model agent state as a dynamic graph, where memories, tools, skills, workflows, and inter-agent relations are represented as typed nodes, edges, and subgraphs updated through schema-constrained rewrites. Based on this formulation, we organize existing dynamic-graph-based methods for self-evolving agents into four taxonomies: node/feature evolution, edge/topology evolution, subgraph activation, and cross-component co-evolution. Building on this taxonomy, we propose dynamic graph learning as reusable infrastructure for self-evolving agents and map nine dynamic-graph-learning subfields to agent-evolution capabilities, discussing their adaptations and possible failure modes. Finally, we discuss five types of graph-aware evaluation and governance protocols from a dynamic-graph perspective, which complement end-task evaluation. The goal is to provide a compact structural lens for designing and governing self-evolving agents.