THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF BOUNDED DEGREE GRAPHS ON SURFACES

Let G be an n-vertex (n ≥ 3) simple graph embeddable on a surface of Euler genus (the number of crosscaps plus twice the number of handles). In this paper, we present upper bounds for the signless Laplacian spectral radius of planar graphs, outerplanar graphs and Halin graphs, respectively, in terms of order and maximum degree. We also demonstrate that our bounds are sometimes better than known ones. For outerplanar graphs without internal triangles, we determine the extremal graphs with the maximum and minimum signless Laplacian spectral radii.

[1]  Charles R. Johnson,et al.  Matrix analysis , 1985, Statistical Inference for Engineers and Data Scientists.

[2]  Jinlong Shu,et al.  Sharp upper bounds on the spectral radius of graphs , 2004 .

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

[4]  B. Mohar.  THE LAPLACIAN SPECTRUM OF GRAPHS y , 1991 .

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

[6]  Hong Yuan.  On the Spectral Radius and the Genus of Graphs , 1995, J. Comb. Theory, Ser. B.

[7]  Maxima of the $Q$-index for outer-planar graphs , 2014, 1405.0392.

[8]  Willem H. Haemers,et al.  Spectra of Graphs , 2011 .

[9]  J. A. Bondy,et al.  Graph Theory with Applications , 1978 .

[10]  Shuchao Li,et al.  On the signless Laplacian spectra of k-trees , 2015, 1507.02536.

[11]  Lilian Markenzon,et al.  Subclasses of k-trees: Characterization and recognition , 2006, Discret. Appl. Math..

[12]  G. Royle,et al.  A Conjecture on the Maximum Value of the Principal Eigenvalue of a Planar Graph , 2010 .

[13]  Dragoš Cvetković,et al.  Towards a spectral theory of graphs based on the signless Laplacian, III , 2010 .

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

[16]  Mark N. Ellingham,et al.  The Spectral Radius of Graphs on Surfaces , 2000, J. Comb. Theory, Ser. B.

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

[18]  D. Cvetkovic,et al.  Towards a spectral theory of graphs based on the signless Laplacian, I , 2009 .

[19]  D. Cao,et al.  The spectral radius of a planar graph , 1993 .

[20]  Zdenek Dvorak,et al.  Spectral radius of finite and infinite planar graphs and of graphs of bounded genus , 2009, J. Comb. Theory, Ser. B.

[21]  D. Cvetkovic,et al.  The largest eigenvalue of a graph: A survey , 1990 .

[22]  Lilian Markenzon,et al.  Generating and counting unlabeled k-path graphs , 2014, Discret. Appl. Math..

[23]  D. Stevanović,et al.  The signless Laplacian spectral radius of graphs on surfaces , 2013 .