Abstract: Firstly, we gave an upper bound for the
vertex-distinguishing proper edge chromatic number of composition of
complete graph $K_{p}$ and star $S_{q}$, which is $pq+1$ for $p\geq
2,~q\geq4$. Then by constructing coloring in terms of the symmetry
of regular polygons and the methods of combinatorial analysis, we
obtained  respectively vertex-distinguishing proper edge chromatic
numbers for composition of complete graph $K_{p}$ and star $S_{q}$
when $p=2,~q\geq4$; $p\geq3,~ q=4$; $p$ is even and $p\geq4,~q=5$;
$pq$ is odd and $p\geq3,~q\geq5$.

Key words: composition, complete graph, star, vertex-distinguishing proper edge coloring, vertex-distinguishing proper edge chromatic number