the eigenvalues and energy of integral circulant graphs

a graph is called textit{circulant} if it is a cayley graph on a cyclic group, i.e. its adjacency matrix is circulant. let $d$ be a set of positive, proper divisors of the integer $n>1$. the integral circulant graph $icg_{n}(d)$ has the vertex set $mathbb{z}_{n}$ and the edge set e$(icg_{n}(d))= {{a,b}; gcd(a-b,n)in d }$. let $n=p_{1}p_{2}cdots p_{k}m$, where $p_{1},p_{2},cdots,p_{k}$ are distinct prime numbers and $gcd(p_{1}p_{2}cdots p_{k},m)=1$. the open problem posed in paper [a. ili'{c}, the energy of unitary cayley graphs, linear algebra appl., 431 (2009) 1881--1889] about calculating the energy of an arbitrary integral circulant $icg_{n}(d)$ is completely solved in this paper, where $d={p_{1},p_{2},ldots,p_{k} } $.