Question

If Qn = (V, E) is the cube graph, how many functions f : V → V

are graph isomorphisms for n = 1, 2, or 3?

Answer 1

In a cube graph Qn, for n=1 there are 2 graph isomorphisms, for n=2 there are 8 graph isomorphisms, and for n=3 there are 48 graph isomorphisms.

Step-by-step explanation:

In graph theory, a cube graph Qn is a graph representing an n-dimensional cube, where each vertex corresponds to a corner of the cube and each edge connects two corners that are at a Hamming distance of 1 from each other. Now, the number of graph isomorphisms for cube graphs differs for different n values.

For n=1, we have a line with two vertices, so the functions f : V → V that are graph isomorphisms are the identities and the function that swaps the two vertices, thus, we have 2 isomorphisms.

For n=2, we have a square (with 4 vertices), so the number of isomorphisms would be all permutations of the vertices that preserve adjacency, which are 8 in total.

For n=3, we have a cube (with 8 vertices). It has 48 graph isomorphisms which can be calculated by counting the number of distinct orientations of a cube in 3D space.