For a simple undirected connected graph G of order n, let , , and 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 is signified by , where Here, we propose a new kind of Estrada index based on the Gaussianization of the generalized distance matrix of a graph. Let be the generalized distance eigenvalues of a graph G. We define the generalized distance Gaussian Estrada index , as Since characterization of is very appealing in quantum information theory, it is interesting to study the quantity and explore some properties like the bounds, the dependence on the graph topology G and the dependence on the parameter . In this paper, we establish some bounds for the generalized distance Gaussian Estrada index of a connected graph G, involving the different graph parameters, including the order n, the Wiener index , the transmission degrees and the parameter , and characterize the extremal graphs attaining these bounds.
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