Maximum Laplacian energy among threshold graphs

Vinagre, Cybele T. M.; Del-Vecchio, Renata R.; Justo, Dagoberto Adriano Rizzotto; Trevisan, Vilmar

doi:10.1016/j.laa.2013.04.030

Cited by 18 publications

(12 citation statements)

References 6 publications

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“…Mais informações sobre os grafos threshold podem ser encontradas em [1], [3] e [4]. Caso o grafo seja conexo, temos a n = 1.…”

Section: Grafos Thresholdunclassified

Grafos split completos com energia laplaciana sem sinal máxima

Pinheiro¹,

Trevisan²

2017

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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Resumo. Neste trabalho descreveremos o grafo split completo com n vértices que tem a maior energia laplaciana sem sinal.Palavras-chave. Energia laplaciana sem sinal, grafos threshold, grafos split completos. IntroduçãoSeja G = (V, E) um grafo simples, onde V = {v 1 , v 2 , . . . , v n }é o conjunto de vértices de G e E = {e 1 , e 2 , . . . , e n }é o conjunto de arestas de G. Usaremos d i para denotar o grau do vértice v i . A matriz laplaciana sem sinal Q de Gé uma matriz simétrica de ordem n × n definida elemento a elemento como q ij = 1, se os vértices v i e v j são adjacentes e 0 caso contrário, se i = j, e q ii = d i . Os autovalores de Q são chamados de autovalores laplacianos sem sinal de G e são denotados por q 1 , q 2 , . . . , q n .Um grafo G com n = k + j vérticesé chamado de split completo se Gé formado por uma clique com j vértices e um conjunto independente de k vértices, onde todos esses k vértices são adjacentes aos j vértices da clique.Definimos a energia laplaciana sem sinal de G porO conceito de energia de grafos surgiu inicialmente relacionadoà matriz de adjacência A de G, onde a energiaé definida como a soma dos valores absolutos dos autovalores de A. Com o aprofundamento do estudo de outras matrizes associadasà grafos (laplaciana, laplaciana normalizada, laplaciana sem sinal, ...) surgiram outros conceitos de energia em grafos, incluindo a que trataremos neste trabalho: energia laplaciana sem sinal. Uma questão relevante que surge quando trabalhamos com energiaé encontrar grafos extremais, por exemplo, qual dentre os grafos com n vértices, possui energia máxima. Seguindo

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“…Mais informações sobre os grafos threshold podem ser encontradas em [1], [3] e [4]. Caso o grafo seja conexo, temos a n = 1.…”

Section: Grafos Thresholdunclassified

Grafos split completos com energia laplaciana sem sinal máxima

Pinheiro¹,

Trevisan²

2017

Proceeding Series of the Brazilian Society of Computational and Applied Mathematics

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“…Here b i is 0 if vertex v i was added as an isolated vertex, and b i is 1 if v i was added as a dominating vertex. This representation has been called a creation sequence [12,13]. For example, the binary sequence of S k,n−k constructed in Fig.2.…”

Section: Characterization Of K-trees With the Laplacian Energymentioning

confidence: 99%

The Laplacian energy and Laplacian eigenvalues of the K-trees

Zhao¹,

Wang²

2016

imf

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A k-tree is either a complete graph on k vertices or a graph obtained from a smaller k-tree by adjoining a new vertex together with k edges connecting it to a k-clique. Denote the set of all n-vertex k-trees by T k n. In this paper, we impose some restrictions on the spectrum of a k-tree with the k number has largest Laplacian energy and smallest Laplacian energy among all the graphs satisfying those conditions. The corresponding extremal graphs are characterized respectively as well.

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“…, n [8]. It has been conjectured that among all connected graphs on n nodes the threshold graph called pineapple with trace ⌊ 2n 3 ⌋ maximizes the Laplacian energy (see [11]). Among connected threshold graphs the pineapple is indeed the maximizer; for general threshold graphs on n nodes the clique of size ⌊ 2n+1 3 ⌋ + 1 together with ⌊ n−3 3 ⌋ isolated vertices is a threshold graph maximizing Laplacian energy [6] and we conjecture that this graph has maximum Laplacian energy among all graphs on n nodes.…”

Section: Introductionmentioning

confidence: 99%

Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy

Helmberg

Trevisan

2017

Linear Algebra and its Applications

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The Laplacian energy of a graph is the sum of the distances of the eigenvalues of the Laplacian matrix of the graph to the graph's average degree. The maximum Laplacian energy over all graphs on n nodes and m edges is conjectured to be attained for threshold graphs. We prove the conjecture to hold for graphs with the property that for each k there is a threshold graph on the same number of nodes and edges whose sum of the k largest Laplacian eigenvalues exceeds that of the k largest Laplacian eigenvalues of the graph. We call such graphs spectrally threshold dominated. These graphs include split graphs and cographs and spectral threshold dominance is preserved by disjoint unions and taking complements. We conjecture that all graphs are spectrally threshold dominated. This conjecture turns out to be equivalent to Brouwer's conjecture concerning a bound on the sum of the k largest Laplacian eigenvalues.

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Maximum Laplacian energy among threshold graphs

Cited by 18 publications

References 6 publications

Grafos split completos com energia laplaciana sem sinal máxima

Grafos split completos com energia laplaciana sem sinal máxima

The Laplacian energy and Laplacian eigenvalues of the K-trees

Spectral threshold dominance, Brouwer's conjecture and maximality of Laplacian energy

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