Automorphism Groups of Rational Circulant Graphs

Klin, Mikhail; Kovács, István

doi:10.37236/2039

Cited by 13 publications

(11 citation statements)

References 93 publications

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“…In case n ≡ 1 mod 2, we have γ n,d = 1, thus max d|n, d< n 2 γ n,d d = max d|n, d< n 2 d = n p 1 , which by use of ( 8) verifies (6) and also shows that D = { n p 1 } is the unique corresponding singleton divisor set for odd n.…”

Section: Proof Theorem 8 Implies Thatmentioning

confidence: 71%

“…A motivation for investigating spectra of graphs is to find out to which extent the eigenvalues of a graph G characterise G. According to a conjecture of So [16] different integral circulant graphs have different spectra, taking multiplicities of eigenvalues into account. In [6,Corollaries 11.2 and 11.3] this was confirmed by Klin and Kovács as a consequence of Zibin's conjecture for arbitrary circulant graphs, which in turn follows from work of Muzychuk [10] on the structure of Schur rings over cyclic groups. Independently, Dobson and Morris [3] proved Toida's conjecture by using the classification of finite simple groups and deduced Zibin's more general conjecture from it.…”

Section: Introductionmentioning

confidence: 73%

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The Geometric Kernel of Integral Circulant Graphs

Sander¹

2021

Electron. J. Combin.

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By a suitable representation in the Euclidean plane, each circulant graph $G$, i.e. a graph with a circulant adjacency matrix ${\mathcal A}(G)$, reveals its rotational symmetry and, as the drawing's most notable feature, a central hole, the so-called \emph{geometric kernel} of $G$. Every integral circulant graph $G$ on $n$ vertices, i.e. satisfying the additional property that all of the eigenvalues of ${\mathcal A}(G)$ are integral, is isomorphic to some graph $\mathrm{ICG}(n,\mathcal{D})$ having vertex set $\mathbb{Z}/n\mathbb{Z}$ and edge set $\{\{a,b\}:\, a,b\in\mathbb{Z}/n\mathbb{Z} ,\, \gcd(a-b,n)\in \mathcal{D}\}$ for a uniquely determined set $\mathcal{D}$ of positive divisors of $n$. A lot of recent research has revolved around the interrelation between graph-theoretical, algebraic and arithmetic properties of such graphs. In this article we examine arithmetic implications imposed on $n$ by a geometric feature, namely the size of the geometric kernel of $\mathrm{ICG}(n,\mathcal{D})$.

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Section: Proof Theorem 8 Implies Thatmentioning

confidence: 71%

Section: Introductionmentioning

confidence: 73%

The Geometric Kernel of Integral Circulant Graphs

Sander¹

2021

Electron. J. Combin.

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“…In this paper, we determine the automorphism group of unitary Cayley graphs X n , and make a step in describing the automorphism group of integral circulant graphs by examining two special cases -n being a prime power or a square-free number [22,27]. Our proofs are based on the fact that for some primes p dividing n, the classes modulo p permute under the automorphism f .…”

Section: Discussionmentioning

confidence: 99%

“…In this paper we characterize the automorphism group Aut(X n ) of unitary Cayley graphs, and make a step towards characterizing the automorphism group of an arbitrary integral circulant graph. Many authors studied the isomorphisms of circulant and Cayley graphs [26,28], automorphism groups of Cayley digraphs [10], integral Cayley graphs over Abelian groups [24], rational circulant graphs [22], etc. For the survey on the automorphism groups of circulant graphs see [27].…”

Section: Introductionmentioning

confidence: 99%

On the Automorphism Group of Integral Circulant Graphs

Bašić¹,

Ilić²

2011

Electron. J. Combin.

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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, 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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“…In [2] David A. Smith discussed the fixed points of automorphisms group of a semi-simple Lie algebra. Several authors deliberate different properties of automorphism groups, see for further [3,7,8,10,11]. In [4], Christoper J. Hillar and Darren L. Rhea found the automorphism group of finite abelian groups.…”

Section: Introductionmentioning

confidence: 99%

Fixed points of automorphisms of certain finite groups

Ali¹,

Hayat²,

Li³

2019

ija

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Let G = Z p 2 × Z p 3 be a finite group of order p 5. Suppose that d is a divisor of the order of group G. In this article, we enumerated all automorphisms of G fixing d elements of G and denote them by Θ(G, d), we characterize the automorphism group as 2 by 2 matrices with entries are from finite groups. We also compute a formula for Θ(Q 4p , d), where Q 4p is the dicyclic group of order 4p, p is an odd prime and the divisor d ∈ {1, 2, 4, p, 2p, 4p}.

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Automorphism Groups of Rational Circulant Graphs

Cited by 13 publications

References 93 publications

The Geometric Kernel of Integral Circulant Graphs

The Geometric Kernel of Integral Circulant Graphs

On the Automorphism Group of Integral Circulant Graphs

Fixed points of automorphisms of certain finite groups

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