H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors

By Liqun Qi. 2013

193 citations recorded by the source corpus.

We propose a simple and natural definition for the Laplacian and the signless Laplacian tensors of a uniform hypergraph. We study their H$^+$-eigenvalues, i.e., H-eigenvalues with nonnegative H-eigenvectors, and H$^{++}$-eigenvalues, i.e., H-eigenvalues with positive H-eigenvectors. We show that each of the Laplacian tensor, the signless Laplacian tensor and the adjacency tensor has at most one H$^{++}$-eigenvalue, but has several other H$^+$-eigenvalues. We identify their largest and smallest H$^+$-eigenvalues, and establish some maximum and minimum properties of these H$^+$-eigenvalues. We then define analytic connectivity of a uniform hypergraph and discuss its application in edge connectivity.

Subfields: Biomaterials, Computational Mathematics, Radiology, Nuclear Medicine and Imaging

View Paper: https://www.semanticscholar.org/paper/47e9f92f130c4140b72a0ff2d50f045b9240e7d7

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In Biomaterials

In Computational Mathematics

In Radiology, Nuclear Medicine and Imaging

Citations in the corpus

Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process (2013, background): “The Laplacian tensor and the signless Laplacian tensor of G are defined by L(G) = D(G)−A(G) and Q(G) = D(G)+A(G), which were initially introduced in [21], and studied further in [11, 13, 22].”
Hankel Tensors: Associated Hankel Matrices and Vandermonde Decomposition (2013, background): “genvalues) are nonnegative. On the other hand, copositive tensors do not restrict the order to be even, thus are more general. Nonnegative tensors, positive semi-deﬁnite tensors and Laplacian tensors [12] are copositive tensors [11]. In the next section, for each Hankel tensor A ∈ H m,n, we associat”
Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues (2013, background): “By the techniques in [9, 20], the conclusions on connected hypergraphs can be easily generalized to general hypergraphs.”
Properties of Tensor Complementarity Problem and Some Classes of Structured Tensors (2014, background): “ymmetric nonnegative tensors and so on. Beside automatical control, positive semi-deﬁnite tensors also found applications in magnetic resonance imaging [25, 26, 27, 28] and spectral hypergraph theory [29, 30, 31]. The following questions are natural. Can we extend the concept of Q-matrices to Qtenso”
Some spectral properties and characterizations of connected odd-bipartite uniform hypergraphs (2014, background): “In [11], it was mentioned that ‘It is a research topic to identify the conditions under which ρ(L) = ρ(Q)’.”
An S-type eigenvalue localization set for tensors (2015, background): “,...,im∈N aii2···imxi2 ···xim (N = {1,2,...,n}) and x[m−1] = (x m−1 1 ,x2 ,...,x m−1 n ) T. If x and λ are all real, then λ is called an H-eigenvalue of A and x an Heigenvector of A associated with λ [20, 21, 27]. Oneofmany practicalapplications ofeigenvalues oftensors isthatonecan identify the posi”
Symmetric nonnegative tensors and copositive tensors (2012, methodology): “(A) = λ max(A). Combining these together, we have (2). ✷ The adjacency tensor of a uniform hypergraph is a nonnegative tensor [4]. The signless Laplacian tensor of a uniform hypergraph, introduced in [15], is also a nonnegative tensor. Cooper and Dutle [4] established (2) for the adjacency tensor of”
Regular Uniform Hypergraphs, $s$-Cycles, $s$-Paths and Their largest Laplacian H-Eigenvalues (2013, background): “By the techniques in [7, 17], the conclusions on connected hypergraphs can be easily generalized to general hypergraphs.”
M-tensors and nonsingular M-tensors (2013, background): “[7,9,8,15]) is an M-tensor, but is not a nonsingular M-tensor.”
Positive Definiteness and Semi-Definiteness of Even Order Symmetric Cauchy Tensors (2014, background): “The following definition is consistent with [1] and [12] respectively.”
An eigenvalue localization set for tensors with applications to determine the positive (semi-)definiteness of tensors (2016, background): “7 Let A = (ai jkl) ∈ R[4,3] be a real symmetric tensor with elements defined as follows: a1111 = 4.”
The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph (2013, background): “By the techniques in [18,10], the conclusions on connected hypergraphs can be easily generalized to general hypergraphs.”
An even order symmetric B tensor is positive definite (2014, background): “agonally dominated symmetric tensor is positive semi-deﬁnite. We will show this in Section 2. Based upon this, we know that the Laplacian tensor in spectral hypergraph theory is positive semi-deﬁnite [9, 10, 11, 18, 23]. Song and Qi [25] showed that an even order Hilbert tensor is positive deﬁnite. ”