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 drawings.
The Wells graph has LCF notation signature ((8, 2), (4, 9), (2, 51), (1, 339848)) (E. Weisstein, Mar. 11, 2026). Of these, there is 1 order-4, 5 order-2, and 554 distinct order-1 LCF notations that give bilaterally symmetric drawings.
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 graph cover of the graph 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 distance-3 graph distance 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 graph 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. 2026) and may be termed the Brussels graph (E. Weisstein, Nov. 6, 2025) and Spence graph (E. Weisstein, Nov. 6, 2025), respectively.
The Wells graph has graph genus 9 (E. Weisstein, Jan. 13, 2026).
The Wells graph is implemented in the Wolfram Language as GraphData["WellsGraph"].
