Spectral Densest Subgraph and Independence Number of a Graph

In this paper, we study spectral versions of the densest subgraph problem and the largest independence subset problem. In the first part, we give an algorithm for identifying small subgraphs with large spectral radius. We also prove a Hoffman-type ratio bound for the order of an induced subgraph whose spectral radius is bounded from above.

[1]  Andries E. Brouwer,et al.  Eigenvalues and perfect matchings , 2005 .

[2]  Jirí Matousek,et al.  Graph coloring with no large monochromatic components , 2007, Electron. Notes Discret. Math..

[3]  Herbert S. Wilf,et al.  Spectral bounds for the clique and independence numbers of graphs , 1986, J. Comb. Theory, Ser. B.

[4]  A. Nilli,et al.  Tight Estimates for Eigenvalues of Regular Graphs , 2004 .

[5]  Michael Doob,et al.  Spectra of graphs , 1980 .

[6]  Yonatan Bilu.  Tales of Hoffman: Three extensions of Hoffman's bound on the graph chromatic number , 2006, J. Comb. Theory, Ser. B.

[7]  A. J. Hoffman,et al.  ON EIGENVALUES AND COLORINGS OF GRAPHS, II , 1970 .

[8]  Vadim V. Lozin,et al.  Maximum k-regular induced subgraphs , 2007, J. Comb. Optim..

[9]  Noga Alon,et al.  Bipartite Subgraphs and the Smallest Eigenvalue , 2000, Combinatorics, Probability and Computing.

[10]  Qiao Li,et al.  Spectral radii of graphs with given chromatic number , 2007, Appl. Math. Lett..

[11]  Jon Kleinberg,et al.  Authoritative sources in a hyperlinked environment , 1999, SODA '98.

[12]  N. Linial,et al.  Expander Graphs and their Applications , 2006 .

[13]  Chris D. Godsil,et al.  Eigenvalue bounds for independent sets , 2008, J. Comb. Theory, Ser. B.

[14]  Vladimir Nikiforov,et al.  Some Inequalities for the Largest Eigenvalue of a Graph , 2002, Combinatorics, Probability and Computing.

[15]  Béla Bollobás,et al.  Cliques and the spectral radius , 2007, J. Comb. Theory, Ser. B.

[16]  W. Haemers.  Interlacing eigenvalues and graphs , 1995 .

[17]  Sebastian M. Cioaba.  On the extreme eigenvalues of regular graphs , 2006, J. Comb. Theory, Ser. B.

[18]  R.E. Kooij,et al.  Robustness of networks against viruses: the role of the spectral radius , 2006, 2006 Symposium on Communications and Vehicular Technology.

[19]  Christos Faloutsos,et al.  Epidemic spreading in real networks: an eigenvalue viewpoint , 2003, 22nd International Symposium on Reliable Distributed Systems, 2003. Proceedings..

[20]  Russell Merris,et al.  The Laplacian Spectrum of a Graph II , 1994, SIAM J. Discret. Math..

[21]  Jirí Matousek,et al.  Graph Colouring with No Large Monochromatic Components , 2008, Comb. Probab. Comput..

[22]  David A. Gregory,et al.  Large matchings from eigenvalues , 2007 .

[23]  Willem H. Haemers,et al.  Perfect matchings in regular graphs from eigenvalues , 2007 .