Resistance characterizations of equiarboreal graphs

A weighted (unweighted) graph G is called equiarboreal if the sum of weights (the number) of spanning trees containing a given edge in G is independent of the choice of edge. In this paper, we give some resistance characterizations of equiarboreal weighted and unweighted graphs, and obtain the necessary and sufficient conditions for k-subdivision graphs, iterated double graphs, line graphs of regular graphs and duals of planar graphs to be equiarboreal. Applying these results, we obtain new infinite families of equiarboreal graphs, including iterated double graphs of 1-walk-regular graphs, line graphs of triangle-free 2-walk-regular graphs, and duals of equiarboreal planar graphs.