On the Laplacian-energy-like invariant

Abstract Let G be a connected graph of order n with Laplacian eigenvalues μ 1 ⩾ μ 2 ⩾ ⋯ ⩾ μ n - 1 > μ n = 0 . The Laplacian-energy-like invariant of the graph G is defined as LEL = LEL ( G ) = ∑ i = 1 n - 1 μ i . Lower and upper bounds for LEL are obtained, in terms of n, number of edges, maximum vertex degree, and number of spanning trees.

[1]  I. Gutman.  Estimating the Laplacian Energy-Like Molecular Structure Descriptor , 2012 .

[2]  Ernesto Estrada.  Spectral theory of networks : from biomolecular to ecological systems , 2009 .

[3]  Xing Gao,et al.  Resistance distances and the Kirchhoff index in Cayley graphs , 2011, Discret. Appl. Math..

[4]  Bao-Xuan Zhu.  The Laplacian-energy like of graphs , 2011, Appl. Math. Lett..

[5]  I. Gutman,et al.  Resistance distance and Laplacian spectrum , 2003 .

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

[7]  I. Gutman,et al.  Laplacian energy of a graph , 2006 .

[8]  I. Gutman.  The Energy of a Graph: Old and New Results , 2001 .

[9]  Fuji Zhang,et al.  Clique-inserted-graphs and spectral dynamics of clique-inserting , 2009 .

[10]  A. Mohammadian,et al.  Graphs with four distinct Laplacian eigenvalues , 2011 .

[11]  F. Kittaneh,et al.  Improved Young and Heinz inequalities for matrices , 2010 .

[12]  Bolian Liu,et al.  A Laplacian-energy-like invariant of a graph , 2008 .

[13]  J. Palacios,et al.  Broder and Karlin's formula for hitting times and the Kirchhoff Index , 2011 .

[15]  Dariush Kiani,et al.  On incidence energy of a graph , 2009 .

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

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

[18]  Bolian Liu,et al.  A Survey on the Laplacian-Energy-Like Invariant ∗ , 2011 .

[19]  Bojan Mohar,et al.  The Quasi-Wiener and the Kirchhoff Indices Coincide , 1996, J. Chem. Inf. Comput. Sci..

[20]  Kinkar Chandra Das,et al.  A Sharp Upper Bound for the Number of Spanning Trees of a Graph , 2007, Graphs Comb..

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

[22]  Ernesto Estrada,et al.  Topological atomic displacements, Kirchhoff and Wiener indices of molecules , 2009, 0912.2628.

[23]  Shigeru Furuichi,et al.  On refined Young inequalities and reverse inequalities , 2010, 1001.0535.

[24]  Aleksandar Ilic,et al.  ON LAPLACIAN LIKE ENERGY OF TREES , 2011, 1103.4814.

[25]  Bo Zhou,et al.  More Upper Bounds for the Incidence Energy , 2010 .

[26]  R. Merris.  A survey of graph laplacians , 1995 .

[27]  Bo Zhou,et al.  The Laplacian-energy Like Invariant is an Energy Like Invariant , 2010 .

[28]  H. Kober.  On the arithmetic and geometric means and on Hölder’s inequality , 1958 .

[29]  New Results on the Incidence Energy of Graphs 1 , 2012 .

[30]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[31]  José Luis Palacios.  Resistance distance in graphs and random walks , 2001 .

[32]  Piotr Biler,et al.  Problems in mathematical analysis , 1990 .

[33]  Weizhong Wang.  Asymptotic behavior of Laplacian-energy-like invariant of some graphs , 2014, Ars Comb..

[34]  K. Das,et al.  EXTREMAL LAPLACIAN-ENERGY-LIKE INVARIANT OF GRAPHS WITH GIVEN MATCHING NUMBER ∗ , 2013 .

[35]  Douglas J. Klein,et al.  Molecular cyclicity and centricity of polycyclic graphs. I. Cyclicity based on resistance distances or reciprocal distances , 1994 .

[36]  Kexiang Xu,et al.  Comparison between Kirchhoff index and the Laplacian-energy-like invariant , 2012 .

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

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

[39]  Guang-Hui Xu,et al.  3-Regular digraphs with optimum skew energy , 2012 .

[40]  Yanfeng Luo,et al.  On Laplacian-energy-like invariant of a graph < , 2012 .

[41]  Douglas J. Klein,et al.  Resistance-Distance Sum Rules* , 2002 .