## 4 Upper Bounds for the Generalized Distance Estrada Index of Graphs
** Starting with Equation (2), we have  
\[D_{\alpha}E(G) = \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{k} \tag{16}\]6  
Taking into account the definition of the generalized distance energy, we obtain  
\[D_{\alpha}E(G) = \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{k} \tag{16}\]7which leads to  
\[D_{\alpha}E(G) = \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{k} \tag{16}\]8 \[D_{\alpha}E(G) = \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{k} \tag{16}\]9 \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]0  
Notice that the function \(f(x)=e^{x}-x\) monotonically increases for \(x\geq 0\). Therefore, the minimum upper bound for \(D_{\alpha}E(G)-E^{D_{\alpha}}(G)\) is attained for \(2(1-\alpha)^{2}\sum_{1\leq i<j\leq n}d_{ij}^{2}+\alpha^{2}\sum_{i=1}^{n} \operatorname{Tr}_{i}^{2}-\frac{4\alpha^{2}W^{2}(G)}{n}\leq 2(1-\alpha)^{2} \frac{n(n-1)}{2}d^{2}+\frac{\alpha^{2}n^{3}(n-1)^{2}}{4}-\alpha^{2}n(n-1)^{2}\), and we have  
\[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]1 \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]2  
Another way to obtain the relation between \(D_{\alpha}E(G)\) and \(E^{D_{\alpha}}(G)\) is as follows,  
\[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]3 \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]4 \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]5 \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]6  
implying  
\[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]7  
Moreover, equality holds in (21) and (22) if and only if \(G\cong K_{1}\). \(\square\)  
**Theorem 9**. : _Assume that \(G\) is connected with order \(n\) and \(0\leq\alpha<1\). Then,_  
\[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]8  
_where the equality holds if and only if \(G=K_{1}\)._  
**Proof. ** Notice that it holds \(\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)=0\) and \(\partial_{1}\geq\frac{2W(G)}{n}\). From the definition of \(D_{\alpha}E(G)\), we get  
\[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\sum_{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|^{k}\]9 \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\left(\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{2} \right)^{\frac{k}{2}}\]0 \[\leq \sum_{k=0}^{k_{0}}\frac{U_{k}(G)}{k!}+\sum_{k\geq k_{0}+1}\frac{1 }{k!}\left(\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{2} \right)^{\frac{k}{2}}\]1  
with equality attained if and only if \(\sum_{i=1}^{n}\left(\partial_{i}-\frac{2\alpha W(G)}{n}\right)^{k}=\left(\sum _{i=1}^{n}\left|\partial_{i}-\frac{2\alpha W(G)}{n}\right|\right)^{k}\) holds for all integers \(k\geq 2\), i.e., if and only if at most one of \(\partial_{i}-\frac{2\alpha W(G)}{n}\) for \(i=1,2,\ldots,n\) is positive and all others are equal to zero, i.e., \(\partial_{2}=\cdots=\partial_{n}=\frac{2\alpha W(G)}{n}\).