Kernels on graphs have had limited options for node-level problems. To
address this, we present a novel, generalized kernel for graphs with node
feature data for semi-supervised learning. The kernel is derived from a
regularization framework by treating the graph and feature data as two Hilbert
spaces. We also show how numerous kernel-based models on graphs are instances
of our design. A kernel defined this way has transductive properties, and this
leads to improved ability to learn on fewer training points, as well as better
handling of highly non-Euclidean data. We demonstrate these advantages using
synthetic data where the distribution of the whole graph can inform the pattern
of the labels. Finally, by utilizing a flexible polynomial of the graph
Laplacian within the kernel, the model also performed effectively in
semi-supervised classification on graphs of various levels of homophily