Title: DEEP GRAPH SPECTRAL EVOLUTION NETWORKS FOR GRAPH TOPOLOGICAL TRANSFORMATION. Abstract: Characterizing the underlying mechanism of graph topological evolution from a source graph to a target graph has attracted fast increasing attention in the deep graph learning domain. However, there lacks expressive and efficient that can handle global and local evolution patterns between source and target graphs. On the other hand, graph topological evolution has been investigated in the graph signal processing domain historically, but it involves intensive labors to manually determine suitable prescribed spectral models and prohibitive difficulty to fit their potential combinations and compositions. To address these challenges, this paper proposes the deep Graph Spectral Evolution Network (GSEN) for modeling the graph topology evolution problem by the composition of newly-developed generalized graph kernels. GSEN can effectively fit a wide range of existing graph kernels and their combinations and compositions with the theoretical guarantee and experimental verification. GSEN has outstanding efficiency in terms of time complexity ($O(n)$) and parameter complexity ($O(1)$), where $n$ is the number of nodes of the graph. Extensive experiments on multiple synthetic and real-world datasets have demonstrated outstanding performance.