The Spectral Radius and Domination Number of Uniform Hypergraphs

This paper investigates the spectral radius and signless Laplacian spectral radius of linear uniform hypergraphs. A dominating set in a hypergraph H is a subset D of vertices if for every vertex v not in D there exists \(u\in D\) such that u and v are contained in a hyperedge of H. The minimum cardinality of a dominating set of H is called the domination number of H. We give lower bounds on the spectral radius and signless Laplacian spectral radius of a linear uniform hypergraph in terms of its domination number.

[1]  C. Brand,et al.  Eigenvalues and domination in graphs , 1996 .

[2]  Peter J. Slater,et al.  Fundamentals of domination in graphs , 1998, Pure and applied mathematics.

[3]  S. Hedetniemi,et al.  Domination in graphs : advanced topics , 1998 .

[4]  Lek-Heng Lim,et al.  Singular values and eigenvalues of tensors: a variational approach , 2005, 1st IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2005..

[5]  Liqun Qi,et al.  Eigenvalues of a real supersymmetric tensor , 2005, J. Symb. Comput..

[6]  V. Nikiforov.  Bounds on graph eigenvalues I , 2006, math/0602027.

[7]  Kung-Ching Chang,et al.  Perron-Frobenius theorem for nonnegative tensors , 2008 .

[8]  Bibin K. Jose,et al.  HYPERGRAPH DOMINATION AND STRONG INDEPENDENCE , 2009 .

[9]  P. Hansen,et al.  A sharp upper bound on algebraic connectivity using domination number , 2009 .

[10]  Qingzhi Yang,et al.  Further Results for Perron-Frobenius Theorem for Nonnegative Tensors , 2010, SIAM J. Matrix Anal. Appl..

[11]  Joshua N. Cooper,et al.  Spectra of Uniform Hypergraphs , 2011, 1106.4856.

[12]  Yuning Yang,et al.  On some properties of nonnegative weakly irreducible tensors , 2011, 1111.0713.

[13]  Csilla Bujtás,et al.  Transversals and domination in uniform hypergraphs , 2012, Eur. J. Comb..

[14]  Michael A. Henning,et al.  Hypergraphs with large domination number and with edge sizes at least three , 2012, Discret. Appl. Math..

[15]  J. Shao.  A general product of tensors with applications , 2012, 1212.1535.

[16]  Bo Zhou,et al.  Laplacian and signless Laplacian spectral radii of graphs with fixed domination number , 2013, 1310.7308.

[17]  S. Gaubert,et al.  Perron–Frobenius theorem for nonnegative multilinear forms and extensions , 2009, 0905.1626.

[18]  Tan Zhang,et al.  A survey on the spectral theory of nonnegative tensors , 2013, Numer. Linear Algebra Appl..

[19]  L. Qi,et al.  Cored Hypergraphs, Power Hypergraphs and Their Laplacian H-Eigenvalues , 2013, 1304.6839.

[20]  L. Qi,et al.  The largest Laplacian and signless Laplacian H-eigenvalues of a uniform hypergraph , 2013, 1304.1315.

[21]  Jan Har.  A note on Laplacian eigenvalues and domination , 2014 .

[22]  Linyuan Lu,et al.  Connected Hypergraphs with Small Spectral Radius , 2014, 1402.5402.

[23]  Lizhu Sun,et al.  Some Spectral Properties of Uniform Hypergraphs , 2014, Electron. J. Comb..

[24]  Juntong Liu.  Generalizations of the Brunn–Minkowski inequality , 2016 .

[25]  Erfang Shan,et al.  Matching and domination numbers in r-uniform hypergraphs , 2017, J. Comb. Optim..

[26]  B. D. Acharya.  Domination in Hypergraphs , 2020 .