A comparative analysis of new graph distance measures and graph edit distance

The problem of determining the structural similarity or distance between graphs got considerable attention during the last decades; consequently, various similarity and distance measures for networks have already been investigated. Dehmer et al. studied a new distance measure for graphs, which is based on topological indices. An interesting problem is to compare our class of measures with the well-known graph edit distance, which has been studied extensively and often serves as a benchmark measure; but note that GED is generally NP-hard. In this paper, we compare the edit distance with our graph distance measures; in order to do so we use some well-known topological indices such as graph energy, Wiener index, Randić index and graph entropy. By using several special graph classes numerical results reveal that the graph distance measure based on graph energy approximates GED well. This fact could trigger an important research direction for studying graph edit distance and other comparative network measures.