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.
Highlights
In this paper, we study connected simple graphs with a finite number of vertices
We give an expression for Pα ( G ) of a regular graph G in terms of the distance eigenvalues as well as adjacency eigenvalues of G, and describe the generalized distance Gaussian Estrada index of some graphs obtained by operations
We present some useful bounds for the generalized distance Gaussian Estrada index Pα ( G ) of a connected graph G, in terms of the different graph parameters including the order n, the Wiener index W ( G ), the transmission degrees and the parameter α ∈ [0, 1]
Summary
We study connected simple graphs with a finite number of vertices. Standard graph terminology will be adopted. D Q ( G ), D1 ( G ) = Tr ( G ) and Dα ( G ) − Dβ ( G ) = (α − β) D L ( G ), any result regarding the spectral properties of generalized distance matrix has its counterpart for each of these particular graph matrices, and these counterparts follow immediately from a straightforward proof. We will refer to ∂1 ( G ) as ∂( G ) in the sequel It follows from the Perron-Frobenius theorem and the non-negativity and irreducibility of Dα ( G ) that ∂( G ) is the unique eigenvalue and there is a unique positive unit eigenvector X corresponding to ∂( G ), which is called the generalized distance Perron vector of G. Kn , Ks,t , Pn and Cn denote, respectively, the complete graph on n vertices, the complete bipartite graph on s + t vertices, the path on n vertices and the cycle on n vertices
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