The Spence graph is the name given in this work to the second cospectral graph of the Wells graph (E. Weisstein, Nov. 8,
2025), i.e., the graph with graph spectrum 
that is not the Wells graph or Brussels
graph.
The graph is so named because E. Spence (private communication reported in van Dam and Haemers 2003) found exactly three graphs with the spectrum of the Wells graph by an exhaustive computer search.
The spence graph has at least 4 LCF emebddings of order 8, 32 of order 4, and 710 or order 2.
The Spence graph satisfies the rhombus constraints and contains no known unit-distance forbidden subgraph, yet appears not to be a unit-distance. A number of embeddings found from different initial embeddings by minimizing the sum of square deviations from unit edge lengths until a local minimum was reached are illustrated above.
