## 1 Introduction
There exists a vast literature related to Estrada index and its bounds and we refer the reader to the nice surveys [15; 16]. This graph-spectrum-based invariant has also an important role in chemistry, biology, as well as network science. It has been applied for instance to gauge the extent of folding of long chain polymeric molecules, encompassing some proteins [8; 17; 18]. It has found a number of applications in complex networks and characterizes the centrality [9]. \(EE\) offers a unique metric to characterize the robustness of complex networks [19]; namely, it is a monotonic measure with respect to the edge deletion and addition. For more applications of the Estrada index in network science, see the monograph in [20] and papers [19; 21]. The pioneering papers [8; 9] further investigate varied versions of Estrada index with respect to other graph associated matrices. Because of the evident success of the graph Estrada index, this proposal has been put into effect, and Estrada index-based on the eigenvalues of more graph matrices have been introduced subsequently: Estrada index-based invariant with respect to distance matrix [22], Estrada index-based invariant with respect to Laplacian matrix [23; 24], Estrada index-based invariant with respect to signless Laplacian matrix [25; 26], and Estrada index-based invariant with respect to distance signless Laplacian matrix [27] have been introduced and studied. For some other interesting papers, we direct the reader to works [28; 29; 30; 31]. The distance matrix graph has shown its importance in a wide range of areas across science and engineering. The distance matrix contains information on the number of walks and self-avoiding walks of chemical graphs, which adjacency matrix fails to show. Distance matrix has been used to calculate many topological indices, including the Wiener index, and thermodynamic properties involving temperature and pressure coefficients. In the design of communication infrastructure network, molecular stability, and graph embedding theory, distance matrix has played an important role. Distance eigenvalues of graphs have attracted huge attention for mathematicians for decades. For more information regarding distance spectrum, we refer to the survey [1] and the references therein. We observe that the distance matrix and its eigenvalues are of importance not only from a chemistry point of view, but also because they are very useful in other branches of science and social science. Clearly, the information one gets regarding the graph from the distance matrix is also visible from the distance Laplacian and the distance signless Laplacian matrices of a graph. As these matrices use more structural properties of a graph than the distance matrix, these matrices may contain a wealth of information about the graph. In addition to its formal analogy to the Estrada index, the distance and the distance signless Laplacian Estrada indices are arguably of prominent significance in physical chemistry. Distance turns out to be rooted deeply in the molecular graphs. Therefore, here we define the _generalized distance Estrada index_\(D_{\alpha}E(G)\), based on the generalized distance matrix of the graph \(G\), as  
\[D_{\alpha}E(G)=\sum_{i=1}^{n}e^{\left(\partial_{i}-\frac{2\alpha W(G)}{n} \right)}, \tag{1}\]  
where \(\partial_{1}\geq\partial_{2}\geq\cdots\geq\partial_{n}\) are the eigenvalues of \(D_{\alpha}(G)\) (the generalized distance eigenvalues of a graph \(G\)). It follows from the definition that \(D_{0}E(G)=DEE(G)\) and \(2D_{1/2}E(G)=D^{Q}EE(G)\), where \(DEE(G)\) denotes the distance Estrada index of a graph \(G\) and \(D^{Q}EE(G)\) denotes the distance signless Laplacian Estrada index of a graph \(G\). This shows that the concept of generalized distance Estrada index of a graph \(G\) merges the theories of distance Estrada index and the distance signless Laplacian Estrada index of a graph \(G\).