The Wells graph, sometimes also called the Armanios-Wells graph, is a quintic graph on 32 nodes and 80 edges that is the unique distance-regular
graph with intersection array 
. It is also distance-transitive.
It is illustrated above in a number of non-LCF embeddings.
The Wells graph possesses at least 2 order-8 LCF embeddings, 9 of order 4, and 3 bilaterally symmetric of order 2, as illustrated above.
It has graph diameter 4, girth 5, graph radius 4, and is Hamiltonian and nonplanar. It has chromatic number 4, edge connectivity 5, and vertex connectivity 5.
The Wells graph can be constructed as follows (Brouwer et al. 1989, p. 266). Label the vertices of a dodecahedral graph
with ordered pairs 
with 
and 
such that the vertex labeled 
is at graph distance 3
from the vertices labeled 
and 
for all 
with 
. Now add 12 new vertices labeled 
and 
for 
, ..., 5 and join 
to all 
, 
to 
, and 
to 
(with 
). The resulting graph is the Wells graph.
The Wells graph is a double cover of the complement of the Clebsch graph (Brouwer et al. 1989, p. 266), i.e., of the halved
cube graph 
.
The Wells graph is also its own graph distance-3 graph.
It has graph spectrum 
(van Dam and Haemers
2003).
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. These graphs have each been rediscovered independently for different reasons (Araujo-Pardo and Leemans 2022; Cambie et al. 2025) and may be termed the Brussels graph (E. Weisstein, Nov. 6, 2025) and Spence graph (E. Weisstein, Nov. 6, 2025), respectively.
The Wells graph is implemented in the Wolfram Language as GraphData["WellsGraph"].
