Co-authorship, rational Erdos numbers, and resistance distances in graphs

The Erdos number (EN) for collaborative papers among mathematicians was defined as indicating the topological distance in the graph depicting the co-authorship relations, i. e., EN = 1 for all co-authors of Paul Erdos; EN = 2 for their co-authors who did not publish jointly with Erdos; etc. A refinement of this notion uses resistance distances leading to rational Erdos numbers (REN), which (as indicated by their name) are rational numbers. For acyclic graphs, EN = REN, but for graphs with circuits these numbers differ. Further refinements are possible using weighted edges in the co-authorship graph according to the number of jointly authored papers.