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 vertices.
It is illustrated above in one of its 108 degree-3 LCF
notations, none of which are bilaterally symmetric.
The Meringer graph is implemented in the Wolfram Language as GraphData["MeringerGraph"].
The Meringer graph has 75 edges, girth 5, graph diameter 3, chromatic
number 3, and is a quintic graph. The group
order of its automorphism group is
96. The graph spectrum of the Meringer graph is
.
The plots above show the adjacency matrix, incidence matrix, and graph distance matrix of the graph.
The Meringer graph satisfies the rhombus constraints and contains no known forbidden subgraph for unit-distance embeddings, yet appears not to be a unit-distance graph. A number of drawings found from different initial drawings by minimizing the sum of squared deviations from unit edge lengths until a local minimum was reached are illustrated above.
