Graph similarity and distance in graphs

Abstract

For connected graphs G1 and G2 of order n and a one-to-one mapping \( \phi : V (G_1) \to V (G_2) \), the \( \phi \)-distance between G1 and G2 is ¶¶ $ d_ \phi (G_1, G_2) = \sum \bigl| d_{G_1} (u,v) - d_{G_2} (\phi u, \phi v)\bigr|, $¶¶ where the sum is taken over all \( {n \choose 2} \) unordered pairs u,v of vertices of G1. The distance between G1 and G2 is defined by \( d (G_1, G_2) = \min \{ d_\phi(G_1, G_2)\} \), where the minimum is taken over all one-to-one mappings \( \phi \) from V (G1) to V (G2). It is shown that this distance is a metric on the space of connected graphs of a fixed order. The maximum distance \( D (G_1, G_2) = \max \{ d_\phi(G_1, G_2)\} \) for connected graphs G1 and G2 of the same order is also explored. The total distance of a connected graph G is \( \Sigma d(u,v) \), where the sum is taken over all unordered pairs u, v of vertices of G. Bounds for d (G1, G2) are presented both in terms of the total distances of G1 and G2 and also in terms of the sizes of G1, G2, and a greatest common subgraph of G1 and G2. For a set \( \cal {S} \) of connected graphs having fixed order, the distance graph \( \cal {D} (\cal {S}) \) of \( \cal {S} \) is that graph whose vertex set is \( \cal {S} \) and such that two vertices G1 and G2 of \( \cal {D}( \cal {S}) \) are adjacent if and only if d (G1, G2) = 1. Furthermore, a graph G is a distance graph if there exists a set \( \cal {S} \) of graphs having fixed order such that \( \cal {D} (\cal {S}) \cong G \). It is shown that every distance graph is bipartite and, moreover, that all even cycles and all forests are distance graphs. Other bipartite graphs are shown to be distance graphs and it is conjectured that all bipartite graphs are distance graphs.