Graph Powers and Graph Homomorphisms

In this paper we investigate some basic properties of fractional powers. In this regard, we show that for any rational number $1\leq {2r+1\over 2s+1}< og(G)$,$G^{{2r+1\over 2s+1}}\longrightarrow H$ifandonlyif$G\longrightarrow H^{-{2s+1\over 2r+1}}.$Also,fortworationalnumbers${2r+1\over 2s+1} < {2p+1\over 2q+1}$andanon-bipartitegraph$G$, we show that $G^{\frac{2r+1}{2s+1}}<G^{\frac{2p+1}{2q+1}}$. In the sequel, we introduce an equivalent definition for circular chromatic number of graphs in terms of fractional powers. We also present a sufficient condition for equality of chromatic number and circular chromatic number.