The Hoffman graph is the bipartite graph on 16 nodes and 32 edges illustrated above that is cospectral
to the tesseract graph 
(Hoffman 1963, van Dam and Haemers 2003). 
and the Hoffman graph are therefore not determined
by their spectrum. Its girth, graph
diameter, graph spectrum, and characteristic
polynomial are the same as those of 
, but its graph radius is
3 compared to the value 4 for 
.
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) |
The Hoffman graph is an integral graph with graph spectrum 
.
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.
