The Clebsch graph, also known as the Greenwood-Gleason graph (Bondy and Murty 1976, p. 242; Read and Wilson, 1998, p. 284) and illustrated above in a number
of drawings, 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 as being distance-transitive.
It can be obtained from the 5-hypercube graph 
by merging antipodal vertices
(i.e., those at graph 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 vertices.
Note that Brouwer et al. (1989, pp. 104 and 224) confusingly use the
term "Clebsch graph" to refer to the 5-halved
cube graph, which is strongly regular
on parameters 
.
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. Bilateral LCF notation drawings 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. The Clebsch graph
is an integral graph and has graph
spectrum 
.
The bipartite double graph of the Clebsch
graph is the hypercube graph 
.
The Clebsch graph appears to have graph crossing number 32. A straight line drawing with 32 crossings was found by Exoo (left figure above), and a bilaterally symmetric straight line drawing with 32 crossings was subsequently found by E. Weisstein (Oct. 20, 2025; right figure).
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.
Folded 5-Cube." 