Ordering uniform supertrees by their spectral radii

Abstract

A supertree is a connected and acyclic hypergraph. For a hypergraph H,the maximal modulus of the eigenvalues of its adjacency tensor is called the spectral radius of H.By applying the operation of moving edges on hypergraphs and the weighted incidence matrix method, we determine the ninth and the tenth k-uniform supertrees with the largest spectral radii among all k-uniform supertrees on nvertices, which extends the known result.

Keywords

Uniform hypergraph / adjacency tensor / uniform supertree / spectral radius

References

[1] BergeC. Hypergraph: Combinatorics of Finite sets.Amsterdam: Elsevier, 1973

[2] ChangA, HuangQ. Ordering trees by their largest eigenvalues.Linear Algebra Appl, 2003, 370: 175–184

[3] ChangK-C, PearsonK, ZhangT. Perron-Frobenius theorem for nonnegative tensors.Commun Math Sci, 2008, 6: 507–520

[4] CooperJ, DutleA. Spectra of uniform hypergraphs.Linear Algebra Appl, 2012, 436: 3268–3299

[5] FriedlandS, GaubertA, HanL. Perron-Frobenius theorems for nonnegative multilinear forms and extensions.Linear Algebra Appl, 2013, 438: 738–749

[6] HofmeisterM. On the two largest eigenvalues of trees.Linear Algebra Appl, 1997, 260: 43–59

[7] HuS, QiL, ShaoJ. Cored hypergraphs, power hypergraphs and their Laplacian eigenvalues.Linear Algebra Appl, 2013, 439: 2980–2998

[8] LiH, ShaoJ, QiL. The extremal spectral radii of k-uniform supertrees.J Comb Optim, 2016, 32: 741–764

[9] LuL, ManS. Connected hypergraphs with small spectral radius.Linear Algebra Appl, 2016, 509: 206–227

[10] QiL. Eigenvalues of a real supersymmetric tensor.J Symbolic Comput, 2005, 40: 1302–1324

[11] YangY, YangQ. On some properties of nonnegative weakly irreducible tensors.arXiv: 1111.0713v2

[12] YuanX, ShaoJ, ShanH. Ordering of some uniform supertrees with larger spectral radii.Linear Algebra Appl, 2016, 495: 206–222

[13] ZhouJ, SunL, WangW, BuC. Some spectral properties of uniform hypergraphs.Electron J Combin, 2014, 21(4): P4.24

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