Spectral Extremal Problems for Hypergraphs

Keevash, Peter; Lenz, John; Mubayi, Dhruv

doi:10.1137/130929370

Cited by 46 publications

(52 citation statements)

References 23 publications

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“…Adjacency matrix. The traditional definition of the adjacency (hyper)matrix (see, e.g., [9] and [11]) represents edges by 1. This tradition was challenged by Cooper and Dutle in [2], who chose to represent the edges of an r-graph by the value 1/ (r − 1)!, thereby scaling down all eigenvalues and simplifying a number of expressions.…”

Section: Discussionmentioning

confidence: 99%

Hypergraphs and hypermatrices with symmetric spectrum

Nikiforov

2017

Linear Algebra and its Applications

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It is well known that a graph is bipartite if and only if the spectrum of its adjacency matrix is symmetric. In the present paper, this assertion is dissected into three separate matrix results of wider scope, which are extended also to hypermatrices. To this end the concept of bipartiteness is generalized by a new monotone property of cubical hypermatrices, called odd-colorable matrices. It is shown that a nonnegative symmetric r-matrix A has a symmetric spectrum if and only if r is even and A is odd-colorable. This result also solves a problem of Pearson and Zhang about hypergraphs with symmetric spectrum and disproves a conjecture of Zhou, Sun, Wang, and Bu.Separately, similar results are obtained for the H-spectram of hypermatrices.

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Section: Discussionmentioning

confidence: 99%

Hypergraphs and hypermatrices with symmetric spectrum

Nikiforov

2017

Linear Algebra and its Applications

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“…This parameter shows remarkable connections with some graph invariants. For instance, λ (1) (G) is equal to the Lagrangian L G of G, which was defined by Motzkin and Straus [3] and satisfies 2L G − 1 = 1/ω(G), where ω(G) is the clique number of G. Obviously λ (2) (G) is the usual spectral radius, and it can be shown that λ (∞) (G)/2 is equal to the number of edges of G.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

The p-spectral radius of the Laplacian matrix

Borba

Fritscher

Hoppen

et al. 2018

APPL ANAL DISCRETE M

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The p-spectral radius of a graph G = (V, E) with adjacency matrix A is defined as λ (p) (G) = max{x T Ax : x p = 1}. This parameter shows remarkable connections with graph invariants, and has been used to generalize some extremal problems. In this work, we extend this approach to the Laplacian matrix L, and define the p-spectral radius of the Laplacian as µ (p) (G) = max{x T Lx : x p = 1}. We show that µ (p) (G) relates to invariants such as maximum degree and size of a maximum cut. We also show properties of µ (p) (G) as a function of p, and a upper bound on max G : |V (G)|=n µ (p) (G) in terms of n = |V | for p ≥ 2, which is attained if n is even.

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“…. , i r } is an edge of G and s(e) is the weight of e. Definition 2.2 (p-spectral radius [32,29]). When p ≥ 1, the p-spectral radius of G, denoted by λ (p) (G), is defined as…”

Section: )mentioning

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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Spectral Extremal Problems for Hypergraphs

Cited by 46 publications

References 23 publications

Hypergraphs and hypermatrices with symmetric spectrum

Hypergraphs and hypermatrices with symmetric spectrum

The p-spectral radius of the Laplacian matrix

Computing the p-Spectral Radii of Uniform Hypergraphs with Applications

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