The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph

By Shenglong Hu, Liqun Qi, Jinshan Xie. 2013

96 citations recorded by the source corpus.

a b s t r a c t In this paper, we show that the largest Laplacian H-eigen- value of a k-uniform nontrivial hypergraph is strictly larger than the maximum degree when k is even. A tight lower bound for this eigenvalue is given. For a connected even- uniform hypergraph, this lower bound is achieved if and only if it is a hyperstar. However, when k is odd, in certain cases the largest Laplacian H-eigenvalue is equal to the maximum degree, which is a tight lower bound. On the other hand, tight upper and lower bounds for the largest signless Laplacian H-eigenvalue of a k-uniform connected hypergraph are given. For connected k-uniform hypergraphs of fixed number of vertices (respectively fixed maximum degree), the upper (respectively lower) bound of their largest signless Laplacian H-eigenvalues is achieved exactly for the complete hypergraph (respectively the hyperstar). The largest Laplacian H-eigenvalue is always less than or equal to the

Subfields: Computational Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/ef0dd1d584683c8008abe3266053b3394e071735

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In Computational Mathematics

1. Chaoqian Li
2. Yaotang Li
3. J. Shao
4. Liqun Qi
5. Anirban Banerjee
6. Shenglong Hu
7. Yimin Wei
8. Weiyang Ding

In Discrete Mathematics and Combinatorics

1. Liqun Qi
2. J. Shao
3. Shenglong Hu
4. Anirban Banerjee

In Geometry and Topology

1. Liqun Qi
2. Anirban Banerjee
3. Chaoqian Li
4. Yaotang Li
5. J. Shao
6. Shenglong Hu
7. Qun Wang

Citations in the corpus

• Double B-tensors and quasi-double B-tensors (2014, background): “B-tensors are linked with positive definite tensors and M-tensors, which are useful in automatical control, magnetic resonance imaging and spectral hypergraph theory [2, 3, 6, 7, 8, 9, 10, 11, 12, 13, 23, 24, 25, 26, 28, 30, 31, 33].”
• On the spectrum of hypergraphs (2017, background): “Different properties of eigenvalues of Laplacian and signless Laplacian tensors of a uniform hypergraph have been studied in [20, 21, 22, 34, 35].”
• M-tensors and nonsingular M-tensors (2013, background): “[7,9,8,15]) is an M-tensor, but is not a nonsingular M-tensor.”
• H$^+$-Eigenvalues of Laplacian and Signless Laplacian Tensors (2013, background): “Some very recent papers [11, 12, 13, 14, 22, 23] demonstrated the impacts on these two aspects.”
• Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues (2013, methodology): “Using the same method as [11], we may generalize an arbitrary graph G to a k-uniform hypergraph Gk , called the kth power of G .”
• Regular Uniform Hypergraphs, $s$-Cycles, $s$-Paths and Their largest Laplacian H-Eigenvalues (2013, background): “The first example for this is the 3-uniform complete hypergraph, given in [9].”
• Some upper bounds on the eigenvalues of uniform hypergraphs (2015, unknown): “(See [5,8].”