Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph

Chang, Jingya; Chen, Yannan; Qi, Liqun

doi:10.1137/16m1060224

Cited by 21 publications

(29 citation statements)

References 55 publications

(65 reference statements)

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“…Also, f (2) = 1 > 0 and f (3) = (−2) k−1 + 1 < 0. Hence, the equation f (λ) = 0 has a unique root in (2,3). That is, (v) holds.…”

Section: Casementioning

confidence: 91%

“…(c) If x k+1 = 0, x 1 = 0 and x 2k = 0, then we have (λ − 2)(λ − 1) k−1 − 1 = 0. By (b2), λ is its root in the interval (0, 1) or (2,3). That is, (ii) holds.…”

Section: Casementioning

confidence: 95%

“…However, [7, Proposition 5.4] can not compute out all H-eigenvalues of its Laplacian tensor. Very recently, Chang, Chen and Qi [3] proposed an efficient first-order optimization algorithm for computing extreme H-and Z-eigenvalues of sparse tensors arising from large scale uniform hypergraphs.These results raised several questions. Firstly, can we identify the largest Laplacian and signless Lapacian H-eigenvalues of k-uniform loose paths?…”

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confidence: 99%

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Computing all Laplacian H-eigenvalues for a uniform loose path of length three

Yue

Zhang

2020

Comp. Appl. Math.

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The spectral theory of Laplacian tensor is an important tool for revealing some important properties of a hypergraph. It is meaningful to compute all Laplacian Heigenvalues for some special k-uniform hypergraphs. For an odd-uniform loose path of length three, the Laplacian H-spectrum has been studied. However, all Laplacian H-eigenvalues of the class of loose paths have not been found out. In this paper, we compute all Laplacian Heigenvalues for the class of loose paths. We show that the number of Laplacian H-eigenvalues of an odd(even)-uniform loose path with length three is 7(14). Some numerical results are given to show the efficiency of our method. Especially, the numerical results show that its Laplacian H-spectrum converges to {0, 1, 1.5, 2} when k goes to infinity. Finally, we establish convergence analysis for a part of the conclusion and also present a conjecture. Recently, Qi, Shao and Wang [20] shown that the largest signless Laplacian H-eigenvalue of a connected k-uniform hypergraph G, reaches its upper bound 2∆, where ∆ is the largest degree of G, if and only if G is regular, and that the largest Laplacian H-eigenvalue of G, reaches the same upper bound, if and only if G is regular and odd-bipartite. What kind of k-uniform hypergraph G is regular and odd-bipartite? To answer this question, they introduced s-paths and s-cycles and studied their properties. Clearly, when s = 1, s-path extended the path in an ordinary graph, which is called loose path [7,14]. They pointed out that s-path cannot be regular but is odd-bipartite when k ≥ 4 [20, Proposition 4.1]. Hu, Qi and Shao [7] introduced the class of cored hypergraphs and power hypergraphs, and investigated the properties of their Laplacian H-eigenvalues. Power hypergraphs are cored hypergraphs, but not vice versa. They showed that loose paths are power hypergraphs, while s-paths and for 2 ≤ s < k 2 are cored hypergraphs, but not power hypergraphs in general. Moreover, it is shown that the largest Laplacian H-eigenvalue of an even-uniform cored hypergraph is equal to its largest signless Laplacian H-eigenvalue. Especially, they found out the Laplacian H-spectra of the k-uniform loose path of length 3 in [7, Proposition 5.4] when k is odd. However, [7, Proposition 5.4] can not compute out all H-eigenvalues of its Laplacian tensor. Very recently, Chang, Chen and Qi [3] proposed an efficient first-order optimization algorithm for computing extreme H-and Z-eigenvalues of sparse tensors arising from large scale uniform hypergraphs.These results raised several questions. Firstly, can we identify the largest Laplacian and signless Lapacian H-eigenvalues of k-uniform loose paths? This question was studied by Yue, Zhang and Lu [22]. For k-uniform loose paths, they showed in [22] that the largest H-eigenvalues of their adjacency tensors, Laplacian tensors, and signless Laplacian tensors are computable. Secondly, can we identify the largest adjacency and signless Lapacian Heigenvalues of power hypergraphs and cored hypergraphs? This question was discussed by Y...

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“…Also, f (2) = 1 > 0 and f (3) = (−2) k−1 + 1 < 0. Hence, the equation f (λ) = 0 has a unique root in (2,3). That is, (v) holds.…”

Section: Casementioning

confidence: 91%

“…(c) If x k+1 = 0, x 1 = 0 and x 2k = 0, then we have (λ − 2)(λ − 1) k−1 − 1 = 0. By (b2), λ is its root in the interval (0, 1) or (2,3). That is, (ii) holds.…”

Section: Casementioning

confidence: 95%

mentioning

confidence: 99%

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Computing all Laplacian H-eigenvalues for a uniform loose path of length three

Yue

Zhang

2020

Comp. Appl. Math.

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“…We note that there are many numerical methods and algorithms for computing partial (real or extreme) eigenvalues of a general (symmetric) tensor; see e.g. Chang et.al [2] and the references therein. We also note the starlike hypergraph is a power hypergraph G k , which is obtained from a starlike simple graph G by adding k − 2 vertices to each of its edges.…”

Section: Conculusionmentioning

confidence: 99%

A combinatorial method for computing characteristic polynomials of starlike hypergraphs

et al. 2019

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“…• A first-order optimization algorithm CEST [8] which is proposed for eigenvalues of large scale sparse tensors involving even order hypergraphs.…”

Section: Computation Of P-spectral Radii Of Hypergraphsmentioning

confidence: 99%

Computing the p-Spectral Radii of Uniform Hypergraphs with Applications

Chang

Ding

et al. 2017

J Sci Comput

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The p-spectral radius of a uniform hypergraph covers many important concepts, such as Lagrangian and spectral radius of the hypergraph, and is crucial for solving spectral extremal problems of hypergraphs. In this paper, we establish a spherically constrained maximization model and propose a first-order conjugate gradient algorithm to compute the p-spectral radius of a uniform hypergraph (CSRH). By the semialgebraic nature of the adjacency tensor of a uniform hypergraph, CSRH is globally convergent and obtains the global maximizer with a high probability. When computing the spectral radius of the adjacency tensor of a uniform hypergraph, CSRH stands out among existing approaches. Furthermore, CSRH is competent to calculate the p-spectral radius of a hypergraph with millions of vertices and to approximate the Lagrangian of a hypergraph. Finally, we show that the CSRH method is capable of ranking real-world data set based on solutions generated by the p-spectral radius model.

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Computing Eigenvalues of Large Scale Sparse Tensors Arising from a Hypergraph

Cited by 21 publications

References 55 publications

Computing all Laplacian H-eigenvalues for a uniform loose path of length three

Computing all Laplacian H-eigenvalues for a uniform loose path of length three

A combinatorial method for computing characteristic polynomials of starlike hypergraphs

Computing the p-Spectral Radii of Uniform Hypergraphs with Applications

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