Two nonisomorphic graphs that have equal resistance spectra (i.e., multisets of resistance distances) are said to be resistance-equivalent.

All nonisomorphic simple graphs on eight or fewer vertices are determined by their resistance spectra. However, exactly 11 pairs of nonisomorphic graphs with nine vertices are resistance-equivalent, illustrated above (in the illustration, the numbers indicate the graph numbers in McKay's enumeration of 9-vertex graphs), where the second and third of these pairs were found by Baxter (1999b).

The numbers of determined by resistance graphs on nodes for , 2, ... are therefore given by 1, 2, 4, 11, 34, 156, 1044,
12346, 274646, 12005070, ... (OEIS A178944),
and the numbers of graphs not determined by resistance are 0, 0, 0, 0, 0, 0, 0, 0,
22, 98, ... (OEIS A178999).

Rickard (1999a) found the pair of 20-vertex resistance-equivalent graphs illustrated above.

Baxter subsequently conjectured that no nonisomorphic biconnected graphs were resistance-equivalent, a conjecture that was nearly immediately refuted by Rickard (1999b) with the pair
of 60-vertex biconnected graphs obtained as
the doubles of the graphs illustrated above.

Baxter, L. "Counterexamples Wanted--Graph Isomorphism & Resistances." sci.math.research
newsgroup posting. Apr. 22, 1999a.Baxter, L. "Counterexample
Wanted for Graph Isomorphism Conjecture." comp.theory
newsgroup posting. Apr. 26, 1999b.McKay, B. "Simple Graphs."
http://cs.anu.edu.au/~bdm/data/graphs.html.Rickard,
J. "Counterexample Wanted for Graph Isomorphism Conjecture." comp.theory
newsgroup posting. Apr. 23, 1999a.Rickard, J. "Counterexample
Wanted for Graph Isomorphism Conjecture." comp.theory
newsgroup posting. Apr. 23, 1999b.Sloane, N. J. A. Sequences
A178944 and A178999,
in "The On-Line Encyclopedia of Integer Sequences."