Spectral radius of r-uniform supertrees with perfect matchings

Abstract

A supertree is a connected and acyclic hypergraph. The set of r-uniform supertrees with n vertices and the set of r-uniform supertrees with perfect matchings on rk vertices are denoted by T_n and T_r,k, respectively. H. Li, J. Shao, and L. Qi [J. Comb. Optim., 2016, 32(3): 741–764] proved that the hyperstar Sn,r attains uniquely the maximum spectral radius in T_n. Focusing on the spectral radius in T_r,k, this paper will give the maximum value in T_r,k and their corresponding supertree.

Keywords

Supertrees / spectral radius / perfect matching

References

[1] Berge C. Hypergraph: Combinatorics of Finite Sets. 3rd ed. Amsterdam: North-Holland, 1973

[2] Bretto A. Hypergraph Theory: An Introduction. Berlin: Springer, 2013

[3] Chang A. On the largest eigenvalue of a tree with perfect matchings. Discrete Math, 2003, 269: 45–63

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

[5] Cvetković D, Doob M, Sachs H. Spectra of Graph|Theory and Applications. New York: Academic Press, 1980

[6] Guo J M, Tan SW. On the spectral radius of trees. Linear Algebra Appl, 2001, 329: 1–8

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

[8] Li H, Shao J, Qi L. The extremal spectral radii of k-uniform supertrees. J Comb Optim, 2016, 32(3): 741–764

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

[10] Lin H, Zhou B, Mo B. Upper bounds for H- and Z-spectral radii of uniform hyper-graphs. Linear Algebra Appl, 2016, 510: 205–221

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

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

[13] Schwenk A J, Wilson R J. Eigenvalues of graphs. In: Beineke L W, Wilson R J, eds. Selected Topics in Graph Theory. New York: Academic Press, 1978

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

[15] Xu G H. On the spectral radius of trees with perfect matchings. In: Combinatorics and Graph Theory. Singapore: World Scientic, 1997

[16] Yuan X. Ordering uniform supertrees by their spectral radii. Front Math China, 2017, 12(6): 1–16

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

[18] Yuan X, Zhang M, Lu M. Some upper bounds on the eigenvalues of uniform hyper-graphs. Linear Algebra Appl, 2015, 484: 540{549

[19] Zhang W, Kang L, Shan E, Bai Y. The spectra of uniform hypertrees. Linear Algebra Appl, 2017, 533: 84–94

[20] Zhou J, Sun L, Wang W, Bu C. Some spectral properties of uniform hypergraphs, Electron J Combin, 2014, 21(4): 4–24

RIGHTS & PERMISSIONS

Higher Education Press and Springer-Verlag GmbH Germany, part of Springer Nature