We introduce a new class of graph kernels based on tropical geometry and the topology of metric graphs. Unlike traditional graph kernels that are defined by graph combinatorics (nodes, edges, subgraphs), our approach considers only the geometry and topology of the underlying metric space. A key property of our construction is its invariance under edge subdivision, making the kernels intrinsically well-suited for comparing graphs that represent different underlying spaces. Our kernels are efficient to compute and depend only on the graph genus rather than the size. In label-free settings, our kernels outperforms existing methods, which we showcase on synthetic, benchmarking, and real-world data and real-world road data. Joint work with Yueqi Cao.