On the Laplacian spectral radius of bipartite graphs with fixed order and size

Abstract Let G n , m be the set of all connected bipartite graphs of order n and size m . In this paper, the problem on maximum Laplacian spectral radius of graphs in G n , m is considered. Among G n , m with n ⩽ m ⩽ 2 n − 5 , the largest Laplacian spectral radius of graphs is determined. As well the upper bound on Laplacian spectral radius of graphs among G n , l ( n − l ) with 2 ⩽ l ⩽ ⌊ n 2 ⌋ is determined. All the corresponding extremal graphs are characterized, respectively.

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

[3]  Dragoš Cvetković,et al.  EIGENVALUE BOUNDS FOR THE SIGNLESS LAPLACIAN , 2007 .

[4]  B. Mohar.  Some applications of Laplace eigenvalues of graphs , 1997 .

[5]  Ying Liu,et al.  Some results on the ordering of the Laplacian spectral radii of unicyclic graphs , 2008, Discret. Appl. Math..

[6]  Ivan Gutman,et al.  THE PATH IS THE TREE WITH SMALLEST GREATEST LAPLACIAN EIGENVALUE , 2002 .

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

[8]  S. Simic,et al.  On bounds for the index of double nested graphs , 2011 .

[9]  R. Brualdi.  Spectra of digraphs , 2010 .

[11]  Zoran Stanic,et al.  Sharp spectral inequalities for connected bipartite graphs with maximal Q-index , 2013, Ars Math. Contemp..

[12]  Shu-Guang Guo.  Ordering trees with n vertices and matching number q by their largest Laplacian eigenvalues , 2008, Discret. Math..

[13]  Lihua Feng,et al.  ON THREE CONJECTURES INVOLVING THE SIGNLESS LAPLACIAN SPECTRAL RADIUS OF GRAPHS , 2009 .

[14]  Jia-Yu Shao,et al.  On the Laplacian spectral radii of bicyclic graphs , 2008, Discret. Math..

[15]  Shuchao Li,et al.  Extremal Halin graphs with respect to the signless Laplacian spectra , 2016, Discret. Appl. Math..

[16]  Shmuel Friedland,et al.  On the First Eigenvalue of Bipartite Graphs , 2008, Electron. J. Comb..

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

[18]  Shuchao Li,et al.  On ordering bicyclic graphs with respect to the Laplacian spectral radius , 2011, Appl. Math. Lett..

[19]  Vladimir Nikiforov,et al.  A Spectral Erdős–Stone–Bollobás Theorem , 2007, Combinatorics, Probability and Computing.

[20]  The Laplacian spectral radius of some bipartite graphs , 2008 .

[21]  D. Cvetkovic,et al.  Graphs for which the least eigenvalue is minimal, II , 2008 .

[22]  Ji-Ming Guo.  On the Laplacian spectral radius of a tree , 2003 .