Graph neural network (GNN) is a framework for directly characterizing graph structured data by deep learning, and has caught increasing attention in recent years. However, the traditional GNN based on message passing aggregation (MP-GNN) ignores the smoothing speed of different nodes and aggregates the neighbor information indiscriminately, which is prone to the over-smoothing phenomenon. Thus, this study proposes a graph kernel neural network classification method KENN based on linear structural entropy. KENN firstly adopts the graph kernel method to encode node subgraph structure, determines isomorphism among subgraphs, and then utilizes the isomorphism coefficient to define the smoothing coefficient among different neighbors. Secondly, it extracts the graph structural information based on the low-complexity linear structural entropy to deepen and enrich the structural expression capability of the graph data. This study puts forward a graph kernel neural network classification method by deeply integrating linear structural entropy, graph kernel and GNN, which can solve the sparse node features of biomolecular data and information redundancy generated by leveraging node degree as features in social network data. It also enables the GNN to adaptively adjust its ability to characterize the graph structural features and makes GNN beyond the upper bound of MP-GNN (WL test). Finally, experiments on seven public graph classification datasets verify that the proposed model outperforms other benchmark models.