Nordhaus–Gaddum type inequalities for the two largest Laplacian eigenvalues

Grijó, Rodrigo; Lima, Leonardo de; Oliveira, Carla; Porto, Guilherme; Trevisan, Vilmar

doi:10.1016/j.dam.2019.04.005

Cited by 5 publications

(3 citation statements)

References 16 publications

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“…Also, we can take the graphs where f (µ 2 , µ 2 , n) = 0 and extend the conjecture presenting the extremal graphs. After running RioGraphX for all graphs ranging from 5 to 10 vertices, no counterexamples were found and the obtained extremal graphs motivated the statement of Conjecture 5 in [Grijó et al, 2019]. Table 5 shows the results of the tests with different numbers of workers (nodes) where we can see that the proposed conjecture was confirmed in times considered optimal given the size of the dataset used (see Table 4), a large number of calculations performed and the ordering of the results.…”

Section: Experimental Evaluationmentioning

confidence: 63%

A Science Gateway to Support Research in Spectral Graph Theory

Oliveira

Abreu²,

Ogasawara³

et al. 2019

Anais Do XXXIV Simpósio Brasileiro De Banco De Dados (SBBD 2019)

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Describing classes of graphs that optimize a function of the eigenvalues subject to some constraints is one of the topics addressed by Spectral Graph Theory (SGT). In this paper, we propose RioGraphX, a science gateway developed on top of Apache Spark, which aims to obtain all graphs that optimize a given mathematical function of the eigenvalues of a graph. Initial experiments involving small graphs have pointed out optimal graphs in a reasonable computational time, and also have shown that leveraging parallel processing is a promising approach to handle larger graphs.

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Section: Experimental Evaluationmentioning

confidence: 63%

A Science Gateway to Support Research in Spectral Graph Theory

Oliveira

Abreu²,

Ogasawara³

et al. 2019

Anais Do XXXIV Simpósio Brasileiro De Banco De Dados (SBBD 2019)

Self Cite

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“…Therefore, we have µ n−2 (H(n 3 , n 4 )) = 2. Then by Lemma 5, we have In [12](see Theorem 7), the authors proved that µ n−2 (G) + µ n−2 (G) 󰃍 2 is true for the graph G when D(G) ∕ = 2 and D(G) ∕ = 2. Combining with this result, µ n−2 (G)+µ n−2 (G) 󰃍 2 is proved to be true for all graphs except the graph G which satisfys D(G) = 2, D(G) = 3, and both G and G have a K 3 as a subgraph.…”

Section: Ng-inequality For µ N−2 (G)mentioning

confidence: 98%

“…Furthermore, Grijó et al [12] studied NG-inequality for µ n−2 (G). They showed that µ n−2 (G) + µ n−2 (G) 󰃍 2 when G or G is disconnected, G is a bipartite graph, a regular graph, or when G and G have diameter not equal to 2.…”

Section: Introductionmentioning

confidence: 99%

Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues

Li,

Guo

2024

Electron. J. Combin.

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Let $G$ be a simple connected graph and $\mu_1(G) \geq \mu_2(G) \geq \cdots \geq \mu_n(G)$ be the Laplacian eigenvalues of $G$. Let $\overline{G}$ be the complement of $G$. Einollahzadeh et al.[J. Combin. Theory Ser. B, 151(2021), 235–249] proved that $\mu_{n-1}(G)+\mu_{n-1}(\overline{G})\geq 1$. Grijò et al. [Discrete Appl. Math., 267(2019), 176–183] conjectured that $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ for any graph and proved it to be true for some graphs. In this paper, we prove $\mu_{n-2}(G)+\mu_{n-2}(\overline{G})\geq 2$ is true for some new graphs. Furthermore, we propose a more general conjecture that $\mu_k(G)+\mu_k(\overline{G})\geq n-k$ holds for any graph $G$, with equality if and only if $G$ or $\overline{G}$ is isomorphic to $K_{n-k}\vee H$, where $H$ is a disconnected graph on $k$ vertices and has at least $n-k+1$ connected components. And we prove that it is true for $k\leq \frac{n+1}{2}$, for unicyclic graphs, bicyclic graphs, threshold graphs, bipartite graphs, regular graphs, complete multipartite graphs and c-cyclic graphs when $n\geq 2c+8$.

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Some Nordhaus-Gaddum type results ofAα-eigenvalues of weighted graphs

Wang

et al. 2021

Applied Mathematics and Computation

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Nordhaus–Gaddum type inequalities for the two largest Laplacian eigenvalues

Cited by 5 publications

References 16 publications

A Science Gateway to Support Research in Spectral Graph Theory

A Science Gateway to Support Research in Spectral Graph Theory

Nordhaus-Gaddum Type Inequalities for the $k$th Largest Laplacian Eigenvalues

Some Nordhaus-Gaddum type results ofAα-eigenvalues of weighted graphs

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