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Graph kernels have emerged as a fundamental and widely adopted technique in graph machine learning. However, most existing graph kernel methods rely on fixed graph similarity estimation that cannot be directly optimized for task-specific objectives, leading to sub-optimal performance. To address this limitation, we propose a kernel-based learning framework called Hierarchical Shortest-Path Graph Kernel Network HSP-GKN, which seamlessly integrates graph similarity estimation with downstream tasks within a unified optimization framework. Specifically, we design a hierarchical shortest-path graph kernel that efficiently preserves both the semantic and structural information of a given graph by transforming it into hierarchical features used for subsequent neural network learning. Building upon this kernel, we develop a novel end-to-end learning framework that matches hierarchical graph features with learnable $hidden$ graph features to produce a similarity vector. This similarity vector subsequently serves as the graph embedding for end-to-end training, enabling the neural network to learn task-specific representations. Extensive experimental results demonstrate the effectiveness and superiority of the designed kernel and its corresponding learning framework compared to current competitors.