On Generalized Distance Gaussian Estrada Index of Graphs

Abstract

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.

Icosahedral Polyhedra from D6 Lattice and Danzer’s ABCK Tiling

“…We will see in the next section that the Danzer tiling also restricts the 6 lattice in a domain, where the vectors are determined by the pair of integers ( 1 , 2 ) . These quantities studied here are potentially applicable to many graph indices [18]. Vector of the Polyhedron in…”

“…We conclude that the inflated vectors by τ n in (12) can be obtained by replacing m 1 byḿ 1 and m 2 byḿ 2 . For example, radii of the icosahedra projected by D 6 vectors 2ω 1 , 2ω 5 , 2ω 1 + 2ω 5 , 2ω 1 + 4ω 5 (18) are in proportion to 1, τ, τ 2 and τ 3 , respectively. It will not be difficult to obtain the D 6 image of any general vector in the 3D space in the form of c(τ p , τ q , τ r ) where p, q, r ∈ Z.…”

Remarks on Wiener Index of Bipolar Fuzzy Incidence Graphs

“…The reason why the mathematical theory and application technology of fuzzy systems can be focused by researchers and has been rapidly developed in the past 60 years is mainly due to the extensive and profound application background in human society and production practice [1][2][3][4][5][6]. The development of the mathematical theory of fuzzy systems largely depends on the development of fuzzy engineering technology and application.…”

Improve Some of Bounds for the Randic Estrada Index of Graphs

“…An application of the Estrada index in statistical thermodynamic has also been reported [9]. Mathematical properties of the Estrada index were studied in a number of recent works ( [1], [6], [10], [11], [12], [13], [14], [20]).…”