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{\bf Milan Ba\v si\'c and Aleksandar Ili\'c}
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{\bf On the Automorphism Group of Integral Circulant Graphs}
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The integral circulant graph $X_n (D)$ has the vertex set $Z_n = \{0,
1,\ldots$, $n{-}1\}$ and vertices $a$ and $b$ are adjacent, if and
only if $\gcd(a{-}b$, $n)\in D$, where $D = \{d_1,d_2, \ldots, d_k\}$ is a
set of divisors of $n$. These graphs play an important role in
modeling quantum spin networks supporting the perfect state transfer
and also have applications in chemical graph theory. In this paper,
we deal with the automorphism group of integral circulant graphs and
investigate a problem proposed in [W. Klotz, T. Sander, \textit{Some
properties of unitary Cayley graphs}, Electr. J. Comb. 14 (2007),
\#R45]. We determine the size and the structure of the automorphism
group of the unitary Cayley graph $X_n (1)$ and the disconnected graph
$X_n (d)$. In addition, based on the generalized formula for the
number of common neighbors and the wreath product, we completely
characterize the automorphism groups $Aut (X_n (1, p))$ for $n$ being
a square-free number and $p$ a prime dividing $n$, and $Aut (X_n (1,
p^k))$ for $n$ being a prime power.
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