On the automorphism group of integral circulant graphs
type: Item
instance of: scholarly article
title: On the automorphism group of integral circulant graphs
zbMATH Open document ID: 1217.05165
author: Milan Bašić
author: Aleksandar Ilić
published in: The Electronic Journal of Combinatorics
publication date: 2011-06-01T00:00:00Z
full work available at URL: https://eudml.org/doc/230456
full work available at URL: v18i1p68.html
review text: Summary: The integral circulant graph \(Xn(D)\) has the vertex set \(Zn = {0, 1, 2, . . . , n - 1}\) and vertices \(a\) and \(b\) are adjacent, if and only if \(gcd(a - b, n) \in D\), where \(D = {d1, d2, . . . , dk}\) 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 [\textit{W. Klotz} and \textit{T. Sander}, ``Some properties of unitary Cayley graphs,'' Electron. J. Comb. 14, No. 1, Research paper R45, 12 p. (2007; Zbl 1121.05059)]. We determine the size and the structure of the automorphism group of the unitary Cayley graph \(Xn(1)\) and the disconnected graph \(Xn(d)\). In addition, based on the generalized formula for the number of common neighbors and the wreath product, we completely characterize the automorphism groups \(Aut(Xn(1, p))\) for \(n\) being a square-free number and \(p\) a prime dividing \(n\), and \(Aut(Xn(1, pk))\) for \(n\) being a prime power.
Mathematics Subject Classification ID: 05C60
Mathematics Subject Classification ID: 05C25
zbMATH DE Number: 5902998
zbMATH Keywords: automorphism group of the unitary Cayley graph
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5902998