## 1 Introduction  
In this paper, we are concerned only with simple, finite, connected and undirected graphs with vertex set \(V(G)=\{v_{1},v_{2},\ldots,v_{n}\}\) and edge set \(E(G)\). The _order_ and the _size_ of \(G\) is, respectively, the number of vertices and the number of edges of \(G\). The _degree_ of a vertex \(v\), denoted by \(d_{G}(v)\) (or simply \(d_{v}\)) is the cardinality of the set of vertices adjacent to \(v\) in a graph \(G\). The _distance_ between two vertices \(u,v\in V(G)\), denoted by \(d_{uv}\), represents the number of edges in a shortest path between these two end nodes in \(G\). The _diameter_ of \(G\) is the maximum distance between any pair of vertices of \(G\). The _distance matrix_\(D(G)\) of a graph \(G\) is a square symmetric matrix defined as \(D(G)=\left(d_{uv}\right)_{u,v\in V(G)}\). The _transmission_ of a vertex \(v\), denoted by \(Tr_{G}(v)\), is defined as the sum of the distances from \(v\) to all other vertices in \(G\), in other words, \(Tr_{G}(v)=\sum\limits_{u\in V(G)}d_{uv}\). A graph \(G\) is referred to as _k-transmission regular_ when the matrix \(D(G)\) has the constant row sum equal to \(k\). The _Wiener index_ (also called the _transmission_) of a graph \(G\), denoted by \(W(G)\), is the sum of distances between all unordered pairs of vertices in \(G\). Apparently, \(W(G)=\frac{1}{2}\sum_{v\in V(G)}Tr_{G}(v)\). For any vertex \(v_{i}\in V(G)\), the transmission \(Tr_{G}(v_{i})\) is also called the _transmission degree_, shortly denoted by \(Tr_{i}\), and the sequence \(\{Tr_{1},Tr_{2},\ldots,Tr_{n}\}\) is called the _transmission degree sequence_ of the graph \(G\). Let \(Tr(G)=diag(Tr_{1},Tr_{2},\ldots,Tr_{n})\) be the diagonal matrix of vertex transmissions of \(G\). Following [PERSON] and [PERSON] [1; 2], the _distance Laplacian matrix_ and the _distance signless Laplacian matrix_ of a graph \(G\) are defined, respectively, as \(D^{L}(G)=Tr(G)-D(G)\) and \(D^{Q}(G)=Tr(G)+D(G)\). Recently, the spectral properties of the distance Laplacian matrix, distance matrix, as well as distance signless Laplacian matrix have attracted attention of the many researchers and a large number of papers have been published regarding their spectral properties, like spectral radius, energy, Estrada index, second largest eigenvalue, smallest eigenvalue, etc. For some recent works, we refer to [3; 4; 5] and the references cited therein. Motivated by [6], [PERSON] et al. [7] introduced the _generalized distance matrix_\(D_{\alpha}(G)\) as a convex combinations of \(Tr(G)\) and \(D(G)\), defined as \(D_{\alpha}(G)=\alpha Tr(G)+(1-\alpha)D(G)\), for \(0\leq\alpha\leq 1\). Noting that \(D_{0}(G)=D(G),2D_{\frac{1}{2}}(G)=D^{Q}(G),D_{1}(G)=Tr(G)\), and \(D_{\alpha}(G)-D_{\beta}(G)=(\alpha-\beta)D^{L}(G)\), any result regarding the spectral properties of generalized distance matrix has its counterpart for each of these particular graph matrices, and these counterparts following immediately from a single proof. In fact, this matrix reduces to merging the distance Laplacian spectral, distance spectral, as well as distance signless Laplacian spectral theories. As the matrix \(D_{\alpha}(G)\) is real symmetric, all its eigenvalues are real. Therefore, we can arrange them as \(\partial_{1}\geq\partial_{2}\geq\cdots\geq\partial_{n}\). The largest eigenvalue \(\partial_{1}\) of the matrix \(D_{\alpha}(G)\) is called the _generalized distance spectral radius_ of \(G\) (We will denote \(\partial_{1}(G)\) by \(\partial(G)\)). Based upon some geometric characteristics of biomolecules, [PERSON] [8; 9] investigated an expression taking the form  
\[EE(G)=\sum_{i=1}^{n}e^{\lambda_{i}},\]  
where \(\lambda_{1},\lambda_{2},\ldots,\lambda_{n}\) represent all eigenvalues of the adjacency matrix of a molecular graph \(G\). Analytical studies on this quantity was performed later in [10] and the name \"Estrada index\" was proposed in [11]. The properties of the Estrada index have been intensively explored, see, for example [11; 12; 13; 14].