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Open AccessArticle

On Generalized Distance Gaussian Estrada Index of Graphs

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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.
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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