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>۱‎. ‎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_{۱}p_{۲}\cdots p_{k}m‎, ‎where‎ ‎p_{۱},p_{۲},\cdots,p_{k} are distinct prime numbers and‎ ‎gcd(p_{۱}p_{۲}\cdots p_{k},m)=۱‎. ‎The open problem posed in paper‎ ‎[A‎. ‎Ili\'{c}‎, ‎The energy of unitary Cayley graphs‎, ‎Linear Algebra‎ ‎Appl.‎, ‎۴۳۱ (۲۰۰۹) ۱۸۸۱--۱۸۸۹] about calculating the energy of an‎ ‎arbitrary integral circulant ICG_{n}(D) is completely solved in‎ ‎this paper‎, ‎where D=\{p_{۱},p_{۲},\ldots,p_{k} \}‎.

کلیدواژه ها:

graph ، Integral circulant graph ، Eigenvalue ، energy