Eigenvectors of Laplacian or signless Laplacian of hypergraphs associated with zero eigenvalue

Fan, Yi-Zheng; Wang, Yi; Bao, Yan-Hong; Wan, Jiang-Chao; Li, Min; Zhu, Zhu

doi:10.1016/j.laa.2019.06.001

Cited by 11 publications

(5 citation statements)

References 30 publications

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“…Lemma 4.8. [8,29] Let G be a connected k-uniform hypergraph. Then −ρ(A(G)) is an H-eigenvalue of A(G) if and only if k is even and G is odd-bipartite.…”

Section: Theorem 42 [10]mentioning

confidence: 99%

A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs

Wang,

Fan

2020

Preprint

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An oriented hypergraph is a hypergraph together with an incidence orientation such that each edge-vertex incidence is given a label of +1 or −1. An oriented hypergraph is called incidence balanced if there exists a bipartition of the vertex set such that every edge intersects one part of the bipartition in positively incident vertices with the edge and other part in negatively incident vertices with the edge. In this paper, we investigate the incidence balance of oriented hypergraphs and induced signed hypergraphs by means of the eigenvalues of associated matrices and tensors, and provide a spectral method to characterize the incidence balanced oriented hypergraphs and their induced signed hypergraphs.

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“…Lemma 4.8. [8,29] Let G be a connected k-uniform hypergraph. Then −ρ(A(G)) is an H-eigenvalue of A(G) if and only if k is even and G is odd-bipartite.…”

Section: Theorem 42 [10]mentioning

confidence: 99%

A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs

Wang,

Fan

2020

Preprint

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“…By the following lemma, if G is connected and non-odd-bipartite, then λ min (G) > −ρ(G). Lemma 2.4 ( [23,22,28,12]). Let G be a k-uniform connected hypergraph.…”

Section: Basic Notionsmentioning

confidence: 99%

“…However, an odd-bipartite hypergraph can have more than one odd-bipartition. Fan et al [12] given a explicit formula for the number of odd-bipartition of a hypergraph by the rank of its incidence matrix over Z 2 . So, it seems hard to give examples of non-odd-bipartite hypergraphs.…”

Section: Introductionmentioning

confidence: 99%

Minimal Non-Odd-Transversal Hypergraphs and Minimal Non-Odd-Bipartite Hypergraphs

Fan¹,

Wang²,

Wan³

2020

Electron. J. Combin.

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Among all uniform hypergraphs with even uniformity, the odd-transversal or odd-bipartite hypergraphs are closer to bipartite simple graphs than bipartite hypergraphs from the viewpoint of both structure and spectrum. A hypergraph is called odd-transversal if it contains a subset of the vertex set such that each edge intersects the subset in an odd number of vertices, and it is called minimal non-odd-transversal if it is not odd-transversal but deleting any edge results in an odd-transversal hypergraph. In this paper we give an equivalent characterization of the minimal non-odd-transversal hypergraphs by means of the degrees and the rank of its incidence matrix over $\mathbb{Z}_2$. If a minimal non-odd-transversal hypergraph is uniform, then it has even uniformity, and hence is minimal non-odd-bipartite. We characterize $2$-regular uniform minimal non-odd-bipartite hypergraphs, and give some examples of $d$-regular uniform hypergraphs which are minimal non-odd-bipartite. Finally we give upper bounds for the least H-eigenvalue of the adjacency tensor of minimal non-odd-bipartite hypergraphs.

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“…Shao et al [21] proved that the −ρ(G) is an H-eigenvalue of A(G) if and only if k is even and G is odd-bipartite. Some other equivalent conditions are summarized in [9]. Note that −ρ(G) is an eigenvalue of A(G) if and only if k is even and G is odd-colorable [9].…”

Section: Introductionmentioning

confidence: 99%

“…Some other equivalent conditions are summarized in [9]. Note that −ρ(G) is an eigenvalue of A(G) if and only if k is even and G is odd-colorable [9]. So, there exist odd-colorable but non-odd-bipartite hypergraphs [7,19], for which −ρ(G) is an N-eigenvalue.…”

Section: Introductionmentioning

confidence: 99%

The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices

Fan,

Zhu,

Wang

2020

Preprint

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Let G be a connected hypergraph with even uniformity, which contains cut vertices. Then G is the coalescence of two nontrivial connected sub-hypergraphs (called branches) at a cut vertex. Let A(G) be the adjacency tensor of G. The least H-eigenvalue of A(G) refers to the least real eigenvalue of A(G) associated with a real eigenvector. In this paper we obtain a perturbation result on the least H-eigenvalue of A(G) when a branch of G attached at one vertex is relocated to another vertex, and characterize the unique hypergraph whose least H-eigenvalue attains the minimum among all hypergraphs in a certain class of hypergraphs which contain a fixed connected hypergraph.

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Eigenvectors of Laplacian or signless Laplacian of hypergraphs associated with zero eigenvalue

Cited by 11 publications

References 30 publications

A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs

A spectral method to incidence balance of oriented hypergraphs and induced signed hypergraphs

Minimal Non-Odd-Transversal Hypergraphs and Minimal Non-Odd-Bipartite Hypergraphs

The least H-eigenvalue of adjacency tensor of hypergraphs with cut vertices

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