On Generalized Distance Gaussian Estrada Index of Graphs

For a simple undirected connected graph G of order n, let D ( G ) , D L ( G ) , D Q ( G ) and T r ( G ) be, respectively, the distance matrix, the distance Laplacian matrix, the distance signless Laplacian matrix and the diagonal matrix of the vertex transmissions of G. The generalized distance matrix D α ( G ) is signified by D α ( G ) = α T r ( G ) + ( 1 − α ) D ( G ) , where α ∈ [ 0 , 1 ] . Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let ∂ 1 , ∂ 2 , … , ∂ n be the generalized distance eigenvalues of a graph G. We define the generalized distance Gaussian Estrada index P α ( G ) , as P α ( G ) = ∑ i = 1 n e − ∂ i 2 . Since characterization of P α ( G ) is very appealing in quantum information theory, it is interesting to study the quantity P α ( G ) and explore some properties like the bounds, the dependence on the graph topology G and the dependence on the parameter α . In this paper, we establish some bounds for the generalized distance Gaussian Estrada index P α ( G ) of a connected graph G, involving the different graph parameters, including the order n, the Wiener index W ( G ) , the transmission degrees and the parameter α ∈ [ 0 , 1 ] , and characterize the extremal graphs attaining these bounds.

[1]  J. A. Rodríguez-Velázquez,et al.  Atomic branching in molecules , 2006 .

[2]  Oscar Rojo,et al.  Sharp upper bounds on the distance energies of a graph , 2018 .

[3]  Bo Zhou,et al.  On the distance signless Laplacian spectral radius of graphs , 2014 .

[4]  Juan A. Rodríguez-Velázquez,et al.  On a graph-spectrum-based structure descriptor , 2007 .

[5]  Werner Kutzelnigg,et al.  What I like about Hückel theory , 2007, J. Comput. Chem..

[6]  G. Indulal,et al.  Sharp bounds on the distance spectral radius and the distance energy of graphs , 2009 .

[7]  Yilun Shang,et al.  Biased edge failure in scale-free networks based on natural connectivity , 2012 .

[8]  D. Cvetkovic,et al.  Spectra of graphs : theory and application , 1995 .

[9]  P. Hansen,et al.  Distance spectra of graphs: A survey , 2013 .

[10]  Lin-wang Wang,et al.  Solving Schrödinger’s equation around a desired energy: Application to silicon quantum dots , 1994 .

[11]  Yilun Shang,et al.  Lower Bounds for Gaussian Estrada Index of Graphs , 2018, Symmetry.

[12]  Ernesto Estrada,et al.  The Structure of Complex Networks: Theory and Applications , 2011 .

[13]  Ernesto Estrada Characterization of the amino acid contribution to the folding degree of proteins , 2004, Proteins.

[14]  Ernesto Estrada,et al.  Exploring the "Middle Earth" of Network Spectra via a Gaussian Matrix Function , 2016, Chaos.

[15]  Indulal Gopalapillai,et al.  Distance spectrum of graph compositions , 2009 .

[16]  Ernesto Estrada,et al.  Gaussianization of the spectra of graphs and networks. Theory and applications , 2019, Journal of Mathematical Analysis and Applications.

[17]  Abdollah Alhevaz,et al.  On the Distance Signless Laplacian Spectrum of Graphs , 2019 .

[18]  M. Lewin On nonnegative matrices , 1971 .

[19]  Boris Furtula,et al.  Resolvent Estrada Index - Computational and Mathematical Studies , 2015 .

[20]  A. Güngör,et al.  On the Distance Estrada Index of Graphs , 2009 .

[21]  Yilun Shang,et al.  Bounds of distance Estrada index of graphs , 2015, Ars Comb..

[22]  I. Gutman NOTE ON LAPLACIAN ENERGY OF GRAPHS , 2008 .

[23]  Gui-Xian Tian,et al.  The generalized distance matrix , 2019, Linear Algebra and its Applications.

[24]  Shang Yi-Lun Local Natural Connectivity in Complex Networks , 2011 .

[25]  J. A. Peña,et al.  Estimating the Estrada index , 2007 .

[26]  Ernesto Estrada Characterization of 3D molecular structure , 2000 .

[27]  P. Hansen,et al.  Two Laplacians for the distance matrix of a graph , 2011 .

[28]  Dragoš Cvetković,et al.  Applications of Graph Spectra: an Introduction to the Literature , 2009 .

[29]  V. Nikiforov,et al.  Merging the A- and Q-spectral theories , 2016, 1607.03015.

[30]  Hailiang Zhang,et al.  Distance spectral spread of a graph , 2012, Discret. Appl. Math..

[31]  Yilun Shang,et al.  Distance Estrada index of random graphs , 2015 .

[32]  Ambat Vijayakumar,et al.  On Distance Energy of Graphs , 2010 .

[33]  Somnath Paul,et al.  On the distance signless Laplacian spectral radius and the distance signless Laplacian energy of graphs , 2018, Discret. Math. Algorithms Appl..

[34]  S. K. Ayyaswamy,et al.  Signless Laplacian Estrada Index , 2011 .

[35]  Bo Zhou,et al.  LAPLACIAN ESTRADA INDEX OF TREES , 2011, 1106.3041.