Subgraphs as a Measure of Similarity
How similar can two graphs be? The ultimate positive answer to this question is, of course, when the two graphs are isomorphic. However, how much internal structure can two nonisomorphic graphs share? We show what the answer can look like if the measure of similarity between the two graphs is taken to be the number of isomorphic subgraphs which they share. We see how this notion is related to the internal symmetries of a graph and that therefore, for most graphs, their internal structure forces them to be very dissimilar to other graphs. We also indicate some attempts to find nonisomorphic graphs which are very similar in terms of the common subgraphs which they share. We also point out some issues of computational complexity and some possible applications associated with this measure of graph similarity.
KeywordsGraph similarity Isomorphic subgraphs Graph reconstruction Reconstruction numbers
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