The tesseract graph, commonly denoted 
, is the 4-hypercube graph
and the skeleton of the tesseract.
It is a quartic symmetric graph with girth 4 and graph diameter
4. The automorphism group of the tesseract
is of order 
(Buekenhout and Parker 1998). The figures above show several nice embeddings,
the leftmost of which appears in Coxeter (1973) and a number of which can be found
in Carr and Kocay (1999). The final graph embedding
highlights a bipartition of 
.
The tesseract graph has two distinct generalized LCF notations of order 4, five of order 2, and four of order 1, illustrated above.
The order-4 LCF notations are given by ![[(-7,-3),(3,7),(-3,5),(-5,3)]^4](/https/mathworld.wolfram.com/images/equations/TesseractGraph/Inline4.svg)
and ![[(-5,-3),(3,5),(-5,5),(-5,5),(-5,5)]^4](/https/mathworld.wolfram.com/images/equations/TesseractGraph/Inline5.svg)
.
The tesseract graph has graph crossing number 8, rectilinear crossing number 8, and local crossing number 1, as illustrated in the first two embeddings above. These embeddings are due to E. Pegg, Jr. (pers. comm., Aug. 17, 2026).
The tesseract graph is a unit-distance graph, as illustrated above.
Several voltage graph embeddings of 
are illustrated above.
It has graph spectrum 
, making it an integral
graph. It is cospectral with the Hoffman
graph, so neither graph is determined by
spectrum.
The tesseract graph is isomorphic to the 4-Hadamard graph.
It has cycle polynomial
 |
It is implemented in the Wolfram Language as GraphData["TesseractGraph"].
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