The Meringer graph is one of the four 
-cage graphs, discovered
by Meringer (1999) after it had long been thought that only three such cages existed.
Like the other 
-cages,
the Meringer graph has 30 nodes. It is illustrated above in one of its 108 degree-3
LCF notations, none of which are bilaterally symmetric.
The Meringer graph has 75 edges, girth 5, diameter 3, chromatic number 3, and is a quintic graph. The order
of its automorphism group is 96. The graph
spectrum of the Meringer graph is 
.
The plots above show the adjacency, incidence, and distance matrices of the graph.
The Meringer 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.
