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
where denotes the transpose and is defined by
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).