Dihedral Butterfly Digraph and Its Cayley Graph Representation

In this paper, we present a new extension of the butterfly digraph, which is known as one of the topologies used for interconnection networks. The butterfly digraph was previously generalized from binary to d-ary. We define a new digraph by adding a signed label to each vertex of the d-ary butterfly digraph. We call this digraph the dihedral butterfly digraph and study its properties. Furthermore, we show that this digraph can be represented as a Cayley graph. It is well known that a butterfly digraph can be represented as a Cayley graph on the wreath product of two cyclic groups [1]. We prove that a dihedral butterfly digraph can be represented as a Cayley graph in two ways. key words: butterfly digraph, dihedral butterfly digraph, Cayley graph, wreath product