The signless Laplacian spread

Abstract The signless Laplacian spread of G is defined as SQ ( G ) = μ 1 ( G ) - μ n ( G ) , where μ 1 ( G ) and μ n ( G ) are the maximum and minimum eigenvalues of the signless Laplacian matrix of G , respectively. This paper presents some upper and lower bounds for SQ ( G ) . Moreover, the unique unicyclic graph with maximum signless Laplacian spread among the class of connected unicyclic graphs of order n is determined.

[1]  Jun Zhou,et al.  Maximizing spectral radius of unoriented Laplacian matrix over bicyclic graphs of a given order , 2008 .

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

[3]  Wai Chee Shiu,et al.  Some results on the Laplacian eigenvalues of unicyclic graphs , 2009 .

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

[5]  Miroslav Petrović,et al.  Bicyclic graphs for which the least eigenvalue is minimum , 2009 .

[6]  Bolian Liu,et al.  On the spread of the spectrum of a graph , 2009, Discret. Math..

[7]  David A. Gregory,et al.  The spread of the spectrum of a graph , 2001 .

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

[9]  Muhuo Liu,et al.  Ordering of the signless Laplacian spectral radii of unicyclic graphs , 2011, Australas. J Comb..

[10]  K. Das.  The Laplacian spectrum of a graph , 2004 .

[11]  Yi-Zheng Fan.  Largest eigenvalue of a unicyclic mixed graph , 2004 .

[12]  Robert Grone,et al.  Eigenvalues and the degree sequences of graphs , 1995 .

[13]  Yi-Zheng Fan,et al.  Minimizing the least eigenvalues of unicyclic graphs with application to spectral spread , 2008 .

[14]  Dong Liang,et al.  The Laplacian Spread of a Tree , 2008, Discret. Math. Theor. Comput. Sci..

[15]  Yong-Liang Pan,et al.  Sharp upper bounds for the Laplacian graph eigenvalues , 2002 .

[16]  I. Olkin,et al.  Inequalities: Theory of Majorization and Its Applications , 1980 .

[17]  Yi-Zheng Fan,et al.  Unoriented Laplacian maximizing graphs are degree maximal , 2008 .

[18]  New sharp upper bounds for the first Zagreb index , 2009 .

[19]  D. Cvetkovic,et al.  Spectra of Graphs: Theory and Applications , 1997 .

[20]  David W. Lewis,et al.  Matrix theory , 1991 .

[21]  Dragoš Cvetković,et al.  A sharp lower bound for the least eigenvalue of the signless Laplacian of a non-bipartite graph , 2008 .