  A non-commuting graph of a finite group $G$ is a graph whose vertices are
non-central elements of $G$ and two vertices are adjacent if they don't commute
in $G$. In this paper, we study the non-commuting graph of the group $U_{6n}$
and explore some of its properties including the independent number, clique and
chromatic numbers. Also, the general formula of the resolving polynomial of the
non-commuting graph of the group $U_{6n}$ are provided. Furthermore, we find
the detour index, eccentric connectivity, total eccentricity and independent
polynomials of the graph.
