Study on some symmetric properties of hypercube graphs through a proper edge coloring

Abstract

The hypercube graph is a well-known graph with many known properties and applications in many fields of study. In this thesis, we give a specific vertex labeling and a specific proper n-edge-coloring of the n-dimensional hypercube graph in a suitable way so that, for a prime integer n, we decompose its adjacency matrix A to satisfy A = \sum_{l=0}^{n-1} (P^l)^{-1}BP^l for a block diagonal submatrix B of A and a permutation matrix P of order 2^n. As a matter of fact, for a prime integer n, they give a cycle notation for each element of the subgroup of automorphism group generated by a rotation by 2\pi/n of the hypercube graph. In addition, based on the above results, we could embed the hypercube graph in a 3-dimensional sphere so that the automorphism group acts on the vertex set of an embedment as a group generated by a rotation of 2\pi/n radian.