One must remove at least five edges from the graph in order to obtain a subgraph that has exactly one perfect matching.

This is a unique property of this graph among fullerenes in the sense that, for every other number of vertices of a fullerene, there exists at least one fullerene from which one can remove four edges to obtain a subgraph with a unique perfect matching.

[4] The vertices of the 26-fullerene graph can be labeled with sequences of 12 bits, in such a way that distance in the graph equals half of the Hamming distance between these bitvectors.

This can also be interpreted as an isometric embedding from the graph into a 12-dimensional taxicab geometry.

[2] In 2009, The New York Times published a puzzle involving Hamiltonian paths in this graph, taking advantage of the correspondence between its 26 vertices and the 26 letters of the English alphabet.