  We describe the design of a reproducing kernel suitable for attributed
graphs, in which the similarity between the two graphs is defined based on the
neighborhood information of the graph nodes with the aid of a product graph
formulation. We represent the proposed kernel as the weighted sum of two other
kernels of which one is an R-convolution kernel that processes the attribute
information of the graph and the other is an optimal assignment kernel that
processes label information. They are formulated in such a way that the edges
processed as part of the kernel computation have the same neighborhood
properties and hence the kernel proposed makes a well-defined correspondence
between regions processed in graphs. These concepts are also extended to the
case of the shortest paths. We identified the state-of-the-art kernels that can
be mapped to such a neighborhood preserving framework. We found that the kernel
value of the argument graphs in each iteration of the Weisfeiler-Lehman color
refinement algorithm can be obtained recursively from the product graph
formulated in our method. By incorporating the proposed kernel on support
vector machines we analyzed the real-world data sets and it has shown superior
performance in comparison with that of the other state-of-the-art graph
kernels.
