Tesseract Graph

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 graph is of group order (Buekenhout and Parker 1998). The figures above show several nice drawings, the leftmost of which appears in Coxeter (1973) and a number of which can be found in Carr and Kocay (1999). The final graph drawing 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 and .

The tesseract graph has graph crossing number 8, rectilinear crossing number 8, and local crossing number 1, as illustrated in the first two drawings above. These drawings 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 drawings of are illustrated above.

The tesseract graph is isomorphic to the torus grid graph and is therefore toroidal, as illustrated above. The left-hand drawing shows an graph embedding in a fundamental region whose paired boundary sides are identified to form a torus. The right-hand drawing shows a finite patch of the corresponding periodic lift, illustrating how edges continue across these boundaries and where corresponding vertices in different regions represent the same vertex on the torus.

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"].