Groups all of whose undirected Cayley graphs are integral

Abdollahi, Aliréza; Jazaeri, Mojtaba

doi:10.1016/j.ejc.2013.11.007

Cited by 17 publications

(12 citation statements)

References 13 publications

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“…(Klotz and Sander [9]) The only finite abelian Cayley integral groups are E 2 n × E 3 m and E 2 n × Z m 4 , where m, n ≥ 0. The non-abelian Cayley integral groups were found recently by Abdollahi and Jazaeri (see [2,Theorem 1.1]), and independently by Ahmady et al (see [4,Theorem 4.2]): Theorem 1.2. (Abdollahi and Jazaeri [2]; Ahmadi et al [4]) The only finite non-abelian Cayley integral groups are D 6 , Dic 12 and Q 8 × E 2 n , where n ≥ 0.…”

Section: Introductionmentioning

confidence: 93%

On groups all of whose undirected Cayley graphs of bounded valency are integral

Estélyi¹,

Kovács²

2014

Preprint

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A finite group G is called Cayley integral if all undirected Cayley graphs over G are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell and Mohar in the nonabelian case. In this paper we generalize this class of groups by introducing the class G k of finite groups G for which all graphs Cay(G, S) are integral if |S| ≤ k. It will be proved that G k consists of the Cayley integral groups if k ≥ 6; and the classes G 4 and G 5 are equal, and consist of: (1) the Cayley integral groups, (2) the generalized dicyclic groups Dic(E 3 n × Z 6 ), where n ≥ 1.

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Section: Introductionmentioning

confidence: 93%

On groups all of whose undirected Cayley graphs of bounded valency are integral

Estélyi¹,

Kovács²

2014

Preprint

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“…The analogous result for non-Abelian Γ was determined independently by Abdollahi and Jazaeri [1] and Ahmady et al [4]: if every Cayley graph Cay(Γ, S) over a finite non-Abelian group Γ is integral then Γ ∈ {S 3 , Z 3 Z 4 , Q 8 × Z r 2 }, where r 0.…”

Section: Introductionmentioning

confidence: 94%

“…Their spectra are [4,6,4,5] and [6,16,10,3], and [9,16,19, 0]. The spectra not previously known to be realized by a graph are [3,4,1,6], [3,5,9, 0], [5,4,7,4], [6,12,2,9], [8,10,16,1], [10,14,18,2], [12,28,4,15], [22,28,34,5], and [27, 28, 49, 0]. Of the 12 graphs, 3 appear in the census of Potočnik et al [17,18] but were not recognized as integral.…”

Section: Introductionmentioning

confidence: 99%

Quartic integral Cayley graphs

Minchenko

Wanless

2015

Ars Math. Contemp.

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We give exhaustive lists of connected 4-regular integral Cayley graphs and connected 4-regular integral arc-transitive graphs. An integral graph is a graph for which all eigenvalues are integers. A Cayley graph Cay(Γ, S) for a given group Γ and connection set S ⊂ Γ is the graph with vertex set Γ and with a connected to b if and only if ba −1 ∈ S. Up to isomorphism, we find that there are 32 connected quartic integral Cayley graphs, 17 of which are bipartite. Many of these can be realized in a number of different ways by using non-isomorphic choices for Γ and/or different choices for S. A graph is arc-transitive if its automorphism group acts transitively on the ordered pairs of adjacent vertices. Up to isomorphism, there are 27 quartic integral graphs that are arc-transitive. Of these 27 graphs, 16 are bipartite and 16 are Cayley graphs. By taking quotients of our Cayley or arc-transitive graphs we also find a number of other quartic integral graphs. Overall, we find 9 new spectra that can be realised by bipartite quartic integral graphs.

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“…In [9], Klotz Theorem 2. (Abdollahi and Jazaeri [2]; Ahmadi et al [4]) The only finite non-abelian Cayley integral groups are D 6 , Dic 12 and Q 8 × E 2 n , where n 0.…”

Section: Introductionmentioning

confidence: 99%

On Groups all of whose Undirected Cayley Graphs of Bounded Valency are Integral

Estélyi

Kovács

2014

Electron. J. Combin.

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A finite group $G$ is called Cayley integral if all undirected Cayley graphs over $G$ are integral, i.e., all eigenvalues of the graphs are integers. The Cayley integral groups have been determined by Kloster and Sander in the abelian case, and by Abdollahi and Jazaeri, and independently by Ahmady, Bell and Mohar in the non-abelian case. In this paper we generalize this class of groups by introducing the class $\mathcal{G}_k$ of finite groups $G$ for which all graphs $\mathrm{Cay}(G,S)$ are integral if $|S| \le k$. It will be proved that $\mathcal{G}_k$ consists of the Cayley integral groups if $k \ge 6;$ and the classes $\mathcal{G}_4$ and $\mathcal{G}_5$ are equal, and consist of: (1) the Cayley integral groups, (2) the generalized dicyclic groups $Dic(E_{3^n} \times \mathbb{Z}_6),$ where $n \ge 1$.

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Groups all of whose undirected Cayley graphs are integral

Cited by 17 publications

References 13 publications

On groups all of whose undirected Cayley graphs of bounded valency are integral

On groups all of whose undirected Cayley graphs of bounded valency are integral

Quartic integral Cayley graphs

On Groups all of whose Undirected Cayley Graphs of Bounded Valency are Integral

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