The Clebsch graph, also known as the Greenwood-Gleason graph (Read and Wilson, 1998, p. 284) and illustrated above in a number of embeddings, is a strongly
regular quintic graph on 16 vertices and 40
edges with parameters 
.
In fact, it is the unique strongly regular graph
with these parameters (Godsil and Royle 2001, p. 230). It is also distance-regular
with intersection array 
, as well being distance-transitive.
Note that Brouwer et al. (1989, pp. 104 and 224) confusingly use the term "Clebsch graph" to refer to the halved
5-cube graph, which is strongly regular on parameters 
.
It can be obtained from the 5-hypercube graph 
by merging antipodal points (i.e.,
those at distance equal to the graph diameter of
5 away), making it the 5-folded cube graph. It
can also be obtained from the tesseract graph 
by adding edges connecting the antipodal
points.
In addition to being isomorphic to the 5-folded cube graph, the Clebsch graph is also isomorphic to the 16-cyclotomic graph and 2-Keller graph.
The Clebsch graph is implemented in the Wolfram Language as GraphData["ClebschGraph"].
The Clebsch graph has three distinct LCF notations of order 4, six of order 2, and 21 of order 1. Symmetric embeddings corresponding to the order-3 and 2 are illustrated above.
If a loop is added to each vertex, the resulting adjacency matrix is equivalent to a 2-(16,6,2) block design.
The Clebsch graph is nonplanar and Hamiltonian and has chromatic number 4.
Kalbfleisch and Stanton (1968) showed that in a 3-edge coloring of the complete graph 
without monochromatic triangles, the subgraph induced by the edges of any one color
is isomorphic to the Clebsch graph, as illustrated above.
The Clebsch graph is an integral graph and has graph spectrum 
.
The bipartite double graph of the Clebsch graph is the hypercube graph 
.
Folded 5-Cube." 