ABSTRACT
Groups are necessary for understanding of the symmetry of a graph, and for con-
structing graphs with prescribed symmetry. We outline here the main concepts of
group theory which are used in this book. They are used for constructing vari-
ous graphs. Consider the graph of the cube, illustrated in Figure 8.1. Given a 3-
dimensional cube, it could be rotated in several ways, and the cube would still look
the same. A rotation of the cube can be represented as a permutation of the vertices.
For example, if we imagine an axis of rotation through the front and rear faces of
the cube in Figure 8.1, then a clockwise rotation through π/2 could be represented by the permutation (1, 3, 5, 7)(2, 4, 6, 8), where the parentheses indicate that the vertices move in two cycles of four vertices each: 1 maps to 3, which maps to 5, which
maps to 7, which maps to 1, etc.
