Suppose $G$ is a graph and $e=ab$ is an edge of $G$. For a positive integer $k$, the $G$-necklace of length $k$ (with respect to edge $e$), denoted by $N_k(G)$, is the graph constructed as follows: Take the vertex disjoint union of $k$ copies of $G$, say $Q_1 cup Q_2 cup cdots cup Q_k$, where each $Q_i$ is a copy of $G$, with $e_i=a_ib_i$ be the copy of $e=a b$ in $Q_i$. Add a vertex $u$, delete the edges $e_i$ for $i=1, 2, cdots, k$ and add edges: $ua_1, b_1a_2, b_2a_3, cdots, b_{k-1}a_k, b_ku$. This thesis determines the circular chromatic indexes of $G$-necklaces for $G = K_{2n}$ and $G= K_{m, m}$.(¨£¹q¤l½×¤å²Ä¤»­¶)