  Let $G$ be a finite abelian group written additively with identity $0$, and
  $\Omega$ be an inverse closed generating subset of $G$ such that $0\notin
\Omega$. We say that $ \Omega $ has the property \lq\lq{}$us$\rq\rq{} (unique
summation), whenever for every $0 \neq g\in G$ if there are $s_1,s_2,s_3, s_4
\in \Omega $ such that $s_1+s_2=g=s_3+s_4 $, then we have $\{s_1,s_2 \} =
\{s_3,s_4 \}$. We say that a Cayley graph $\Gamma=Cay(G;\Omega)$ is a
$us$-$Cayley\ graph$, whenever $G$ is an abelian group and the generating
subset $\Omega$ has the property \lq\lq{}$us$\rq\rq{}. In this paper, we show
that if $\Gamma=Cay(G;\Omega)$ is a $us$-$Cayley\ graph$, then
$Aut(\Gamma)=L(G)\rtimes A$, where $L(G)$ is the left regular representation of
$G$ and $A$ is the group of all automorphism groups $\theta$ of the group $G$
such that $\theta(\Omega)=\Omega$. Then, as some applications, we explicitly
determine the automorphism groups of some classes of graphs including
M\"{o}bius ladders and $k$-ary $n$-cubes.
