The Hoffman graph is the bipartite integral graph on 16 vertices and 32 edges
with graph spectrum 
that is cospectral
with the tesseract graph 
(Hoffman 1963, van Dam and Haemers 2003). It is illustrated
above in a number of drawings, the third of which shows a minimal crossing and rectilinear
crossing graph drawing corresponding to graph
crossing number and rectilinear crossing
number 8.
Since the Hoffman graph is constructed to be cospectral with 
,
neither graph is determined by its spectrum.
While the girth, graph diameter,
graph spectrum, and characteristic
polynomial of the Hoffman graph are the same as those of 
, its graph radius is 3 compared
to the value 4 for 
.
The Hoffman graph is regular and Hamiltonian; its bilateral LCF drawings are illustrated above.
The Hoffman graph has adjacency matrix given by
![A=[0 D; D^T 0],](/images/equations/HoffmanGraph/NumberedEquation1.svg) |
where 
denotes the transpose and 
is defined by
![D=[1 1 1 1 0 0 0 0; 1 1 1 0 1 0 0 0; 1 0 0 1 0 1 1 0; 0 1 0 1 0 1 0 1; 0 0 1 1 0 0 1 1; 1 0 0 0 1 1 1 0; 0 1 0 0 1 1 0 1; 0 0 1 0 1 0 1 1].](/images/equations/HoffmanGraph/NumberedEquation2.svg) |
It is the smallest known conformally rigid graph that is not edge-transitive or distance-regular (Steinerberger and Thomas 2024).
The Hoffman graph is also its own graph distance-3 graph.
The Hoffman graph is implemented in the Wolfram Language as GraphData["HoffmanGraph"].
