  Online relevance matching is an essential task of e-commerce product search
to boost the utility of search engines and ensure a smooth user experience.
Previous work adopts either classical relevance matching models or
Transformer-style models to address it. However, they ignore the inherent
bipartite graph structures that are ubiquitous in e-commerce product search
logs and are too inefficient to deploy online. In this paper, we design an
efficient knowledge distillation framework for e-commerce relevance matching to
integrate the respective advantages of Transformer-style models and classical
relevance matching models. Especially for the core student model of the
framework, we propose a novel method using $k$-order relevance modeling. The
experimental results on large-scale real-world data (the size is 6$\sim$174
million) show that the proposed method significantly improves the prediction
accuracy in terms of human relevance judgment. We deploy our method to the
anonymous online search platform. The A/B testing results show that our method
significantly improves 5.7% of UV-value under price sort mode.
