The eigenvalues and energy of integral circulant graphs

Document Type: Research Paper

Author

Amirkabir University

Abstract

‎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 $D=\{p_{1},p_{2},\ldots,p_{k} \}$‎.

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