The 
graph, also known as the 77-graph,
is a strongly regular graph on 77 nodes
related to the Mathieu group 
and to the Witt design.
It is illustrated above in an embedding with 11-fold symmetry due to T. Forbes
(pers. comm., Dec. 28, 2007).
It is distance-regular with intersection array 
.
It is also distance-transitive.
It is an integral graph with graph spectrum 
.
It can be obtained from the Witt design by selecting the 77 vectors of length 7 that contain a given symbol (arbitrarily chosen from 1-23)
then eliminating that symbol from each of these vectors and renumbering. The resulting
set of vectors 
gives the unique size 77 Steiner system 
on points 1 to 22. Now consider as vertices the 77
vectors (
),
with 
adjacent if share no terms.
The resulting graph is the 
graph.
Explicitly, the graph can be constructed by taking the following 77 words as vertices and drawing an edge for each pair of vertices that have no letters in common.
| abcilu | abdfrs | abejop | abgmnq | abhktv | acdghp | aceqrv |
| acfjnt | ackmos | ademtu | adinov | adjklq | aefgik | aehlns |
| afhoqu | aflmpv | agjsuv | aglort | ahijmr | aipqst | aknpru |
| bcdekn | bcfgov | bchjqs | bcmprt | bdgijt | bdhlmo | bdpquv |
| beflqt | beghru | beimsv | bfhinp | bfjkmu | bgklps | bikoqr |
| bjlnrv | bnostu | cdfimq | cdjoru | cdlstv | cefpsu | cegjlm |
| cehiot | cfhklr | cginrs | cgkqtu | chmnuv | cijkpv | clnopq |
| defhjv | degoqs | deilpr | dfglnu | dfkopt | dgkmrv | dhiksu |
| dhnqrt | djmnps | efmnor | egnptv | ehkmpq | eijnqu | ejkrst |
| eklouv | fghmst | fgjpqr | fijlos | firtuv | fknqsv | ghilqv |
| ghjkno | gimopu | hjlptu | hoprsv | iklmnt | jmoqtv | lmqrsu |
The 
graph can also be obtained by vertex
deletion of the neighbors of a point in the Higman-Sims
graph (but is not, as claimed by van Dam and Haemers (2003), the subgraph
induced by the vertex neighbors). Also note that van Dam and Haemers (2003) refer
to the doubly truncated Witt graph as
, calling the 77-vertex graph the
"local Higman-Sims graph."
Graph."