Layout of Embedding Circulant Networks into Linear Hexagons and Phenylenes

Abstract

Circulant network has been used for decades in the design of computer and telecommunication networks due to optimal fault-tolerance and routing capabilities. Further, it has been used in VLSI design and distributed computation. Hexagonal chains are of great importance of theoretical chemistry because they are the natural graph representations of benzenoid hydrocarbons, a great deal of investigations in mathematical chemistry has been developed to hexagonal chains. Hexagonal chains are exclusively constructed by hexagons of length one. Phenylenes are a class of chemical compounds in which carbon atoms form 6 and 4 membered cycles. Graph embedding has been known as a powerful tool for implementation of parallel algorithms or simulation of different interconnection networks. An embedding f of a guest graph G into a host graph H is a bijection on the vertices such that each edge of G is mapped into a path of H. The wirelength (layout) of this embedding is defined to be the sum of the length of the paths corresponding to the edges of G. In this paper we obtain the minimum wirelength of embedding circulant networks into linear hexagonal chains and linear phenylenes. Further we discuss the embedding of faulty circulant networks into linear hexagonal chains and linear phenylenes.