## 4 Upper Bounds for the Generalized Distance Estrada Index of Graphs  
In this section, we obtain some upper bounds for the generalized distance Estrada index \(D_{\alpha}E(G)\) of a connected graph \(G\) involving the order \(n\), the Wiener index \(W(G)\), the transmission degrees as well as the parameter \(\alpha\in[0,1]\). We also characterize the extremal graphs attaining these bounds. The next result gives an upper bound for the generalized distance Estrada index \(D_{\alpha}E(G)\) using the order \(n\), the Wiener index \(W(G)\), the transmission degrees as well as the parameter \(\alpha\). **Theorem 6**: _Suppose that \(G\) is connected with order \(n\). For any integer \(k_{0}\geq 2\),_  
\[D_{\alpha}E(G) \leq n-1-\sqrt{2(1-\alpha)^{2}\sum_{1\leq i<j\leq n}d_{ij}^{2}+\alpha ^{2}\sum_{i=1}^{n}\mbox{Tr}_{i}^{2}-\frac{4\alpha^{2}W^{2}(G)}{n}} \tag{16}\] \[+ \sum_{k=2}^{k_{0}}\frac{U_{k}(G)-\left(\sqrt{2(1-\alpha)^{2}\sum _{1\leq i<j\leq n}d_{ij}^{2}+\alpha^{2}\sum_{i=1}^{n}\mbox{Tr}_{i}^{2}-\frac{4 \alpha^{2}W^{2}(G)}{n}}\right)^{k}}{k!}\] \[+ e^{\sqrt{2(1-\alpha)^{2}\sum_{1\leq i<j\leq n}d_{ij}^{2}+\alpha ^{2}\sum_{i=1}^{n}\mbox{Tr}_{i}^{2}-\frac{4\alpha^{2}W^{2}(G)}{n}}},\]  
_where the equality holds if and only if \(G=K_{1}\)._  
**Proof. ** By definition of \(D_{\alpha}E(G)\), we have  
\[D_{\alpha}E(G) = \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{k} \tag{16}\] \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\] \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\left(\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{2} \right)^{\frac{k}{2}}\] \[= \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{ \left(\sqrt{2(1-\alpha)^{2}\sum_{1\leq i<j\leq n}d_{ij}^{2}+\alpha^{2}\sum_{i =1}^{n}\mbox{Tr}_{i}^{2}-\frac{4\alpha^{2}W^{2}(G)}{n}}\right)^{k}}{k!}\] \[= \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+e^{\sqrt{2(1-\alpha)^{2} \sum_{1\leq i<j\leq n}d_{ij}^{2}+\alpha^{2}\sum_{i=1}^{n}\mbox{Tr}_{i}^{2}- \frac{4\alpha^{2}W^{2}(G)}{n}}}\] \[- \sum_{k=0}^{k_{0}}\frac{\left(\sqrt{2(1-\alpha)^{2}\sum_{1\leq i <j\leq n}d_{ij}^{2}+\alpha^{2}\sum_{i=1}^{n}\mbox{Tr}_{i}^{2}-\frac{4\alpha^{ 2}W^{2}(G)}{n}}\right)^{k}}{k!},\]  
and (16) follows. Thanks to (16), the equality is attained in (16) if and only if \(G\) has no non-zero \(D_{\alpha}\)-eigenvalues, i.e., \(G\cong K_{1}\). The proof is complete. \(\square\)  
**Remark 2**: _Notice that we have_\[U_{k}(G) = \sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{k}\] \[+ \sum_{k=2}^{k_{0}}\frac{U_{k}(G)-\left(\sqrt{2(1-\alpha)^{2}\sum _{1\leq i<j\leq n}d_{ij}^{2}+\alpha^{2}\sum_{i=1}^{n}\mbox{Tr}_{i}^{2}-\frac{4 \alpha^{2}W^{2}(G)}{n}}\right)^{k}}{k!}\]0  
_where the second inequality can be derived from the following fact:_  
_For non-negative integers \(a_{1},a_{2},\ldots,a_{n}\) and integer \(k\geq 2\),_  
\[+ \sum_{k=2}^{k_{0}}\frac{U_{k}(G)-\left(\sqrt{2(1-\alpha)^{2}\sum _{1\leq i<j\leq n}d_{ij}^{2}+\alpha^{2}\sum_{i=1}^{n}\mbox{Tr}_{i}^{2}-\frac{4 \alpha^{2}W^{2}(G)}{n}}\right)^{k}}{k!}\]1  
_Therefore, \(U_{k}(G)-\left(\sqrt{U_{2}(G)}\right)^{k}\leq 0\).