Q-integral complete r-partite graphs

Wang, Ligong; Li, Xueliang; Hoede, C.

doi:10.1016/j.laa.2012.09.009

Cited by 10 publications

(8 citation statements)

References 11 publications

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“…In this section, we shall give a sufficient and necessary condition for complete r-partite graphs to be distance integral. Similar results for integrality of complete r-partite graphs were given in [32] and for Q-integrality of complete r-partite graphs were given in [34].…”

Section: A Sufficient and Necessary Condition For Complete R-partite supporting

confidence: 67%

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Distance integral complete r-partite graphs

Yang

Wang

2015

Filomat

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Let D(G) = (d ij) n×n denote the distance matrix of a connected graph G with order n, where d ij is equal to the distance between vertices v i and v j in G. A graph is called distance integral if all eigenvalues of its distance matrix are integers. In this paper, we investigate distance integral complete r-partite graphs K p 1 ,p 2 ,...,pr = K a 1 •p 1 ,a 2 •p 2 ,...,as•ps and give a sufficient and necessary condition for K a 1 •p 1 ,a 2 •p 2 ,...,as•ps to be distance integral, from which we construct infinitely many new classes of distance integral graphs with s = 1, 2, 3, 4. Finally, we propose two basic open problems for further study.

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Section: A Sufficient and Necessary Condition For Complete R-partite supporting

confidence: 67%

“…The study on integral graphs and Q-integral graphs has drawn many scholars' attentions. Results about them are found in [5, 7, 8, 13-16, 22, 26, 32] and [10,23,29,34], respectively.…”

Section: Introductionmentioning

confidence: 99%

Distance integral complete r-partite graphs

Yang

Wang

2015

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“…We also give new sufficient conditions for a construction of infinite classes of Q-integral (S-integral) complete r -partite graphs to a given Q-integral (S-integral) complete r -partite graph. Using these conditions we construct infinite classes of QLS-integral complete multipartite graphs, which affirmatively answers to questions 4.1 and 4.2 of [10] and also questions 4.1 and 4.2 of [14]. Although concrete examples of QLS-integral complete multipartite graphs…”

Section: Introductionmentioning

confidence: 68%

“…For results on A-integral complete r-partite graphs see for example [4-6, 9, 11-13]. For results on Q-integral complete r-partite graphs see for example [8,14], for results on S-integral complete r-partite graphs see [7,8,10] and for results on L-integral complete r-partite graphs see [15].…”

Section: Introductionmentioning

confidence: 99%

QLS-integrality of complete r-partite graphs

Pokorný

2015

Filomat

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A graph G is called A-integral (L-integral, Q-integral, S-integral) if the spectrum of its adjacency (Laplacian, signless Laplacian, Seidel) matrix consists entirely of integers. In this paper we study connections between the Q-(L,S,A) integral complete multipartite graphs. Moreover, new sufficient conditions for a construction of infinite families of QLS-integral complete r-partite graphs K p 1 ,p 2 ,...,p r = K b 1 •p 1 ,b 2 •p 2 ,...,bs•ps from given QLS-integral r-partite graphs K p 1 ,p 2 ,...,p r = K a 1 •p 1 ,a 2 •p 2 ,...,as•ps are given. Using these conditions new infinite classes of such graphs for s = 4, 5, 6 are constructed, which affirmatively answers to questions proposed by Wang, Zhao and Li in [10, 14]. Finally, we propose open problems for further study.

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“…Up to now, just a few classes of integral graphs have been characterized. For a survey and references about integral graphs see [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. With regard to L-spectrum, the degree maximal graphs [19] and weakly quasi-threshold graphs [20] have been proven to be L-integral, and the L-integral trees, unicyclic and bicyclic graphs [21] have also been identified.…”

Section: On the Laplacian Integral Tricyclic Graphsmentioning

confidence: 99%

On the Laplacian integral tricyclic graphs

Huang

Wen

2014

Linear and Multilinear Algebra

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A graph is called Laplacian integral if all its Laplacian eigenvalues are integers. In this paper, we give an edge subdividing theorem for Laplacian eigenvalues of a graph (Theorem 2.1) and characterize a class of k-cyclic graphs whose algebraic connectivity is less than one. Using these results, we determine all the Laplacian integral tricyclic graphs. Furthermore, we show that all the Laplacian integral tricyclic graphs are determined by their Laplacian spectra.

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Q-integral complete r-partite graphs

Cited by 10 publications

References 11 publications

Distance integral complete r-partite graphs

Distance integral complete r-partite graphs

QLS-integrality of complete r-partite graphs

On the Laplacian integral tricyclic graphs

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