Largest signless Laplacian spectral radius of uniform supertrees with diameter and pendent edges (vertices)

Abstract

Let \mathbb{S}(m; d; k) be the set of k-uniform supertrees with m edges and diameter d; and S₁(m; d; k) be the k-uniform supertree obtained from a loose path u₁; e₁; u₂; e₂,..., u_d; e_d; u_d+1 with length d by attaching m — d edges at vertex u_{\lfloor d/2\rfloor +1}: In this paper, we mainly determine S₁(m; d; k) with the largest signless Laplacian spectral radius in \mathbb{S}(m; d; k) for 3≤d≤m –1: We also determine the supertree with the second largest signless Laplacian spectral radius in \mathbb{S}(m; 3; k): Furthermore, we determine the unique k-uniform supertree with the largest signless Laplacian spectral radius among all k-uniform supertrees with n vertices and pendent edges (vertices).

Keywords

Signless Laplacian spectral radius / supertree / hypertree / diameter / pendent edges / pendent vertices

References

[1] Brouwer A E, Haemers W H. Spectra of Graphs. New York: Springer, 2012

[2] Cooper J, Dudtle A. Spectra of uniform hypergraphs. Linear Algebra Appl, 2012, 436: 3268–3292

[3] Cvertković D, Rowlinson P, Simić S. An Introduction to the Theory of Graph Spectra. London Math Soc Stud Texts, Vol 75. Cambridge: Cambridge Univ Press, 2010

[4] Duan C X, Wang L G, Xiao P,Li X H. The (signless Laplacian) spectral radius (of subgraphs) of uniform hypergraphs. Filomat, 2019, 33: 4733–4745

[5] Guo J M, Shao J Y. On the spectral radius of trees with fixed diameter. Linear Algebra Appl, 2006, 413: 131–147

[6] Guo S G,Xu G H, Chen Y G. The spectral radius of trees with nvertices and diameter d: Adv Math (China), 2005, 6: 683–692 (in Chinese)

[7] Hu S L, Qi L Q, Shao J Y.Cored hypergraphs, power hypergraphs and their Laplacian H-eigenvalues. Linear Algebra Appl, 2013, 439: 2980–2998

[8] Li H F, Zhou J, Bu C J.Principal eigenvectors and spectral radii of uniform hyper- graphs. Linear Algebra Appl, 2018, 544: 273–285

[9] Li H H, Shao J Y, Qi L Q. The extremal spectral radii of k-uniform supertrees. J Comb Optim, 2016, 32: 741–764

[10] Lim L H. Singular values and eigenvalues of tensors: a variational approach. In: Proceedings of the IEEE InternationalWorkshop on Computational Advances in Multi- Sensor Adaptive Processing (CAMSAP 05). 2005, 129–132

[11] Lim L H. Eigenvalues of tensors and some very basic spectral hypergraph theory. In: Matrix Computations and Scientific Computing Seminar, April 16, 2008

[12] Lin H Y, Mo B, Zhou B,Weng W M. Sharp bounds for ordinary and signless Laplacian spectral radii of uniform hypergraphs. Appl Math Comput, 2016, 285: 217–227

[13] Liu L L, Kang L Y, Yuan X Y. On the principal eigenvector of uniform hypergraphs. Linear Algebra Appl, 2016, 511: 430–446

[14] Lu L Y, Man S D. Connected hypergraphs with small spectral radius. Linear Algebra Appl, 2016, 509: 206–227

[15] Ouyang C, Qi L Q, Yuan X Y. The first few unicyclic and bicyclic hypergraphs with largest spectral radii. Linear Algebra Appl, 2017, 527: 141–163

[16] Qi L Q. Eigenvalues of a real supersymmetric tensor. J Symbolic Comput, 2005, 40: 1302–1324

[17] Qi L Q. Symmetric nonnegative tensors and copositive tensors. Linear Algebra Appl, 2013, 439: 228–238

[18] Qi L Q. H⁺-eigenvalues of Laplacian and signless Laplacian tensors. Commun Math Sci, 2014, 12: 1045–1064

[19] Xiao P, Wang L G. The maximum spectral radius of uniform hypergraphs with given number of pendant edges. Linear Multilinear Algebra, 2019, 67: 1392–1403

[20] Xiao P, Wang L G, Du Y F. The first two largest spectral radii of uniform supertrees with given diameter. Linear Algebra Appl, 2018, 536: 103–119

[21] Xiao P, Wang L G, Lu Y. The maximum spectral radii of uniform supertrees with given degree sequences. Linear Algebra Appl, 2017, 523: 33–45

[22] Yuan X Y, Shao J Y,Shan H Y. Ordering of some uniform supertrees with larger spectral radii. Linear Algebra Appl, 2016, 495: 206{222

[23] Yue J J, Zhang L P, Lu M, Qi L Q. The adjacency and signless Laplacian spectral radius of cored hypergraphs and power hypergraphs. J Oper Res Soc China, 2017, 5: 27–43

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Higher Education Press