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).

Key words： 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