EIGENVALUE BOUNDS FOR THE SIGNLESS LAPLACIAN

We extend our previous survey of properties of spectra of signless Laplacians of graphs. Some new bounds for eigenvalues are given, and the main result concerns the graphs whose largest eigenvalue is maximal among the graphs with fixed numbers of vertices and edges. The results are presented in the context of a number of computer-generated conjectures.

[1]  R. Merris.  Laplacian matrices of graphs: a survey , 1994 .

[2]  D. Cvetkovic.  Signless Laplacians and line graphs , 2005 .

[3]  W. Haemers,et al.  Which graphs are determined by their spectrum , 2003 .

[4]  D. Cvetkovic,et al.  Signless Laplacians of finite graphs , 2007 .

[5]  Jinlong Shu,et al.  A sharp upper bound on the largest eigenvalue of the Laplacian matrix of a graph , 2002 .

[6]  Xiao-Dong Zhang,et al.  Sharp upper and lower bounds for largest eigenvalue of the Laplacian matrices of trees , 2005, Discret. Math..

[7]  Peter Lancaster,et al.  The theory of matrices , 1969 .

[8]  Edwin R. Hancock,et al.  Eigenspaces for Graphs , 2002, Int. J. Image Graph..

[9]  Chen Yan,et al.  Properties of spectra of graphs and line graphs , 2002 .

[10]  On the second Laplacian eigenvalues of trees of odd order , 2006 .

[11]  P. Hansen,et al.  Variable Neighborhood Search for Extremal Graphs: IV: Chemical Trees with Extremal Connectivity Index , 1999, Comput. Chem..

[12]  Bojan Mohar,et al.  Laplacian matrices of graphs , 1989 .

[13]  Dragoš Cvetković,et al.  Spectral Generalizations of Line Graphs: On Graphs with Least Eigenvalue -2 , 2004 .

[14]  Ping Zhu,et al.  A Study of Graph Spectra for Comparing Graphs , 2005, BMVC.

[15]  ChenYan.  PROPERTIES OF SPECTRA OF GRAPHS AND LINE GRAPHS , 2002 .

[16]  Madhav Desai,et al.  A characterization of the smallest eigenvalue of a graph , 1994, J. Graph Theory.

[17]  L. Beineke,et al.  Topics in algebraic graph theory , 2004 .

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

[19]  V. Sunder,et al.  The Laplacian spectrum of a graph , 1990 .

[20]  Amir Daneshgar,et al.  Graph homomorphisms and nodal domains , 2006 .

[21]  CONNECTED GRAPHS OF FIXED ORDER AND SIZE WITH MAXIMAL INDEX: STRUCTURAL CONSIDERATIONS , 2006 .

[22]  Mustapha Aouchiche,et al.  Variable neighborhood search for extremal graphs. 16. Some conjectures related to the largest eigenvalue of a graph , 2005, Eur. J. Oper. Res..

[23]  Nair Maria Maia de Abreu,et al.  The characteristic polynomial of the Laplacian of graphs in (a,b)-linear classes , 2002 .

[24]  Willem H. Haemers,et al.  Enumeration of cospectral graphs , 2004, Eur. J. Comb..

[25]  Pierre Hansen,et al.  Variable neighborhood search for extremal graphs: 1 The AutoGraphiX system , 1997, Discret. Math..

[26]  A connection between ordinary and Laplacian spectra of bipartite graphs , 2008 .

[27]  D. Cvetkovic,et al.  Eigenspaces of graphs: Bibliography , 1997 .