Self-organizing maps capable of processing graph structured information are a relatively new concept. This paper describes a novel concept on the processing of graph structured information using the self-organizing map framework which allows the processing of much more general types of graphs, e.g. cyclic graphs, directed graphs. Previous approaches to this problem were limited to the processing of bounded graphs, their computational complexity can grow rapidly with the level of connectivity of the graphs concerned, and are restricted to the processing of positional graphs. The novel concept proposed in this paper, namely, by using the clusters formed in the state space of the self-organizing map to represent the strengths of the activation of the neighboring vertices, rather than as in previous approaches, using the state space of the surrounding vertices to represent such strengths of activations. Such an approach resulted in reduced computational demand, and in allowing the processing of non-positional graphs.