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On Generalized Distance Gaussian Estrada Index of Graphs

1
Faculty of Mathematical Sciences, Shahrood University of Technology, P.O. Box 316-3619995161, Shahrood, Iran
2
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK
*
Author to whom correspondence should be addressed.
Symmetry 2019, 11(10), 1276; https://doi.org/10.3390/sym11101276
Received: 1 September 2019 / Revised: 26 September 2019 / Accepted: 5 October 2019 / Published: 11 October 2019
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.
Keywords: Gaussian Estrada index; generalized distance matrix (spectrum); Wiener index; generalized distance Gaussian Estrada index; transmission regular graph Gaussian Estrada index; generalized distance matrix (spectrum); Wiener index; generalized distance Gaussian Estrada index; transmission regular graph
MDPI and ACS Style

Alhevaz, A.; Baghipur, M.; Shang, Y. On Generalized Distance Gaussian Estrada Index of Graphs. Symmetry 2019, 11, 1276.

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