Concordance Graphs

A graph G = (V, E) withn ≥ 2 vertices in V is a concordance graph if it has the same number of induced three-vertex one-edged subgraphs as induced three-vertex two-edged subgraphs. A concordance graph is proper if it is the comparability graph of a partially ordered set of order dimension 1 or 2. These definitions relate to the fact that the Kendall and Spearman correlation measures between linear orders L and L′ are equal if and only if the comparability graph of (V, L ∩ L′) is a proper concordance graph. It is proved thatG is a concordance graph if and only if (n + 1)∑dv + 12 t = 3∑dv2, where dvis the degree of vertex v andt is the number of induced K3’s in G. This equation has been used to identify all concordance graphs and proper concordance graphs for n ≤ 7. Infinite families of proper concordance graphs are noted for each t in {0, 1,⋯ }, and an infinite family of regular concordance graphs is specified. It remains open as to whether any regular concordance graph that is neither edgeless nor complete is also proper.