Resistance distance

Professor Frank Harary has had a singular influence on graph theory through his own extensive original research, through the training of several researchers who themselves have made many important contributions, and through popularizing work, most significantly his text Graph Theory [1]. Frank Harary has contributed to a number of more advanced specialized texts, like that of Buckley and Harary [2] on graphical distance. What might well be the general lesson from all this work is: the possibilities for the development of the field of graph theory are ever open for innovative ideas. An example exhibiting the openness of the field is found in the charming monograph by Doyle and Snell [3], wherein a novel probabilistic representation and consequent theorems for electric-circuit conduction problems are described. This work evidently builds from some initial mathematical work of Nash-Williams [4] in 1960, although the graph-theoretic study of electric circuits began over 150 years ago with Kirchhoff 's analysis [5], with much modem work in electrical engineering. Presumably, a qualitative new graphical distance would potentially be of great interest. Judging from the book Distances in Graphs by Buckley and Harary [2], earlier work has exclusively concemed one general type of graphical distance, where the distance between two sites of a graph is taken as the minimal sum of (positive) edge weights along a path between the two sites. The simplest "canonical" case takes each edge to be of unit weight, but more general weights should be of relevance in