Measuring Closeness of Graphs-The Hausdorff Distance

We introduce an apparatus to measure closeness or relationship of two given objects. This is a topology-based apparatus that uses graph representations of the compared objects. In more detail, we obtain a metric on the class of all pairwise non-isomorphic connected simple graphs to measure closeness of two such graphs. To obtain such a measure, we use the theory of hyperspaces from topology to introduce the notion of the Hausdorff graph of any graph G. Then, using this new concept of Hausdorff graphs combined with the notion of graph amalgams, we present the Hausdorff distance, which proves to be useful when examining the closeness of any two connected simple graphs. We also present many possible applications of these concepts in various areas.