Integral Quartic Cayley Graphs on Abelian Groups

Abdollahi, Aliréza; Vatandoost, Ebrahim

doi:10.37236/576

Cited by 9 publications

(17 citation statements)

References 17 publications

(17 reference statements)

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“…The direct sum decompositions (ds1) and (ds2) imply dim Z{2`Z1 pX`e; Z{2q{xC L pX`eqy Z{2˘" dim Z{2`Z1 pX; Z{2q{xC L pXqy Z{2˘" ξ and therefore X`e P cd ξ C L , completing the proof of statement (1). As to (2), it suffices to note that the proof of (1) may be repeated to yield a proof of (2), the only change required being to restrict e to be an edge whose addition keeps the graph bipartite and to replace 'CO L´1 ' by 'LA L´1 '.…”

Section: Proofs Of the Auxiliary Resultsmentioning

confidence: 60%

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On prisms, Möbius ladders and the cycle space of dense graphs

Heinig

2014

European Journal of Combinatorics

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Abstract. For a graph X, let f 0 pXq denote its number of vertices, δpXq its minimum degree and Z 1 pX; Z{2q its cycle space in the standard graph-theoretical sense (i.e. 1-dimensional cycle group in the sense of simplicial homology theory with Z{2-coefficients). Call a graph Hamiltongenerated if and only if the set of all Hamilton circuits is a Z{2-generating system for Z 1 pX; Z{2q. The main purpose of this paper is to prove the following: for every γ ą 0 there exists n 0 P Z such that for every graph X with f 0 pXq ě n 0 vertices, (1) if δpXq ě p 1 2`γ qf 0 pXq and f 0 pXq is odd, then X is Hamilton-generated, (2) if δpXq ě p 1 2`γ qf 0 pXq and f 0 pXq is even, then the set of all Hamilton circuits of X generates a codimension-one subspace of Z 1 pX; Z{2q, and the set of all circuits of X having length either f 0 pXq´1 or f 0 pXq generates all of Z 1 pX; Z{2q, (3) if δpXq ě p 1 4`γ qf 0 pXq and X is square bipartite, then X is Hamilton-generated. All these degree-conditions are essentially best-possible. The implications in (1) and (2) give an asymptotic affirmative answer to a special case of an open conjecture which according to [European J. Combin. 4 (1983), no. 3, p. 246] originates with A. Bondy.

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Section: Proofs Of the Auxiliary Resultsmentioning

confidence: 60%

“…As to (1), if the host graph is required to be regular in advance, a still unsettled conjecture of B. Jackson [40, p. 13, l. 17] posits that a d-regular graph with d ě f0pXq´1 2 actually realizes the…”

Section: Introductionmentioning

confidence: 99%

On prisms, Möbius ladders and the cycle space of dense graphs

Heinig

2014

European Journal of Combinatorics

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“…It is known that any cyclic group of prime order is a CIS-group (see [8,Corollary 2.19]). Since these are the only simple abelian groups, it is natural to ask the following question ([4, Question 1.4]): Which finite non-abelian simple groups are CIS-groups?…”

Section: Cayley Integral Simple Groupsmentioning

confidence: 99%

“…finite group Γ is called Cayley simple if the only connected integral Cayley graph on Γ is the complete graph of order |Γ|. This definition was first introduced in[8], where the following question was posed ([8, Question 2.21]): Is any finite simple group, Cayley simple? A negative answer to this question is implied in the following result.…”

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confidence: 99%

Eigenvalues of Cayley graphs

Liu,

Zhou

2018

Preprint

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“…Cayley graphs which are integral were studied by many people (e.g. [1,2,4,7,10,11]). Following [10] we call a group G Cayley integral whenever all undirected Cayley graphs over G are integral.…”

Section: Introductionmentioning

confidence: 99%

Groups all of whose undirected Cayley graphs are integral

Abdollahi¹,

Jazaeri²

2014

European Journal of Combinatorics

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Abstract. Let G be a finite group, S ⊆ G \ {1} be a set such that if a ∈ S, then a −1 ∈ S, where 1 denotes the identity element of G. The undirected Cayley graph Cay(G, S) of G over the set S is the graph whose vertex set is G and two vertices a and b are adjacent whenever ab −1 ∈ S. The adjacency spectrum of a graph is the multiset of all eigenvalues of the adjacency matrix of the graph. A graph is called integral whenever all adjacency spectrum elements are integers. Following Klotz and Sander, we call a group G Cayley integral whenever all undirected Cayley graphs over G are integral. Finite abelian Cayley integral groups are classified by Klotz and Sander as finite abelian groups of exponent dividing 4 or 6. Klotz and Sander have proposed the determination of all non-abelian Cayley integral groups. In this paper we complete the classification of finite Cayley integral groups by proving that finite non-abelian Cayley integral groups are the symmetric group S 3 of degree 3, C 3 ⋊ C 4 and Q 8 × C n 2 for some integer n ≥ 0, where Q 8 is the quaternion group of order 8.

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Integral Quartic Cayley Graphs on Abelian Groups

Abstract: A graph is called integral, if its adjacency eigenvalues are integers. In this paper we determine integral quartic Cayley graphs on finite abelian groups. As a side result we show that there are exactly $27$ connected integral Cayley graphs up to $11$ vertices.

Cited by 9 publications

References 17 publications

On prisms, Möbius ladders and the cycle space of dense graphs

On prisms, Möbius ladders and the cycle space of dense graphs

Eigenvalues of Cayley graphs

Groups all of whose undirected Cayley graphs are integral

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