Graph classification is an important problem in graph mining with various applications in different fields. Kernel methods have been successfully applied to this problem, recently producing promising results. A graph kernel that mostly specifies classification performance has to be defined in order to apply kernel methods to a graph classification problem. Although there are various previously proposed graph kernels, the problem is still worth investigating, as the available kernels are far from perfect. In this paper, we propose a new graph kernel based on a recently proposed concept called edge distance-k graphs. These new graphs are derived from the original graph and have the potential to be used as novel graph descriptors. We propose a method to convert these graphs into a multiset of strings that is further used to compute a kernel for graphs. The proposed graph kernel is then evaluated on various data sets in comparison to a recently proposed group of graph kernels. The results are promising, both in terms of performance and computational requirements.