Bai and Hancock recently proposed a novel information theoretic kernel for graphs, namely the Jensen-Shannon graph kernel. One drawback of their approach is that it cannot be applied to either attributed or labeled graphs. In this paper, we aim to define a new Jensen-Shannon diffusion kernel for attributed graphs. We commence by using a tree-index based label strengthening method on an attributed graph with the objection of strengthening the vertex labels. We compute a label entropy to measure the uncertainty of the strengthened labels. With the required label entropies for a pair of graphs to hand, a new kernel for the pair of graphs can be defined by measuring the Jensen-Shannon divergence between the entropies. As the strengthened label of a vertex corresponds to a sub tree rooted at the vertex, the label entropy of a graph is determined by all sub trees identified by the tree-index algorithm. As a result, unlike most existing graph kernels in the literature which merely enumerate pairs of isomorphic substructures, our method incorporates all the identified sub trees into the computation of the kernel. The new kernel thus overcomes the shortcoming of discarding substructures having no corresponding isomorphic substructures. We explore our kernel on several graph datasets abstracted from bioinformatics databases.