## 3 Lower Bounds for the Generalized Distance Estrada Index of Graphs  
In this section, we obtain some lower bounds for the generalized distance Estrada index \(D_{\alpha}E(G)\) of a connected graph \(G\). These bounds are characterized in terms of the order \(n\), the Wiener index \(W(G)\), the transmission degree and the parameter \(\alpha\in[0,1]\). We also investigate the extremal graph attaining these bounds. Our first result gives a lower bound for the generalized distance Estrada index \(D_{\alpha}E(G)\), in terms of the order \(n\), the transmission degrees and the parameter \(\alpha\). **Theorem 2**. : _Suppose \(G\) is a connected graph of order \(n\). Then,_  
\[D_{\alpha}E(G)\geq 1+\sqrt{(n-1)^{2}+2(1-\alpha)^{2}\sum\limits_{1\leq i<j \leq n}(d_{ij})^{2}+\alpha^{2}\left(\sum\limits_{i=1}^{n}Tr_{i}^{2}-\frac{4W^{ 2}(G)}{n}\right)}, \tag{4}\]  
_with equality if and only if \(G\cong K_{1}\)._  
Proof. : Starting with Equation (1), we have  
\[D_{\alpha}E^{2}(G)=\sum\limits_{i=1}^{n}e^{2\left(\partial_{i}-\frac{2\alpha W (G)}{n}\right)}+2\sum\limits_{i<j}e^{\left(\partial_{i}-\frac{2\alpha W(  
The function  
\[f(x)=\frac{1}{2}\left(x+\sqrt{(x-2n)^{2}+4(4-x)(1-\alpha)^{2}\sum_{1\leq i<j\leq n }(d_{ij})^{2}+2\alpha^{2}(4-x)P}\right),\]  
monotonically decreases for \(x\geq 2\). Consequently, the best lower bound for \(D_{\alpha}E(G)\) is attained for \(r=2\). This gives us the first part of the proof. From the derivation of (4), we observe readily that the equality holds if and only if \(G\) has no non-zero \(D_{\alpha}\)-eigenvalues. Recall that \(G\) is connected. Therefore, this can only happen when \(G\cong K_{1}\). The proof is then complete. \(\square\)  
The following result is an immediate consequence of Theorem 2. Corollary 1: _Assume that \(G\) is a connected graph of order \(n\). We have_  
\[D_{\alpha}E(G)\geq 1+\sqrt{(n-1)(n-1+n(1-\alpha)^{2})},\]  
_where the equality holds if and only if \(G\cong K_{1}\)._  
**Proof. ** As \(\sum_{1\leq i<j\leq n}(d_{ij})^{2}\geq\frac{n(n-1)}{2}\), from the lower bound of Theorem 2, we obtain  
\[D_{\alpha}E(G) \geq 1+\sqrt{(n-1)^{2}+2(1-\alpha)^{2}\sum_{1\leq i<j\leq n}(d_{ij})^{ 2}+\alpha^{2}\left(\sum_{i=1}^{n}\Upsilon_{i}^{2}-\frac{4W^{2}(G)}{n}\right)}\] \[\geq 1+\sqrt{(n-1)^{2}+2(1-\alpha)^{2}\frac{n(n-1)}{2}+\alpha^{2} \left(\sum_{i=1}^{n}\Upsilon_{i}^{2}-\sum_{i=1}^{n}\Upsilon_{i}^{2}\right)}\] \[= 1+\sqrt{(n-1)(n-1+n(1-\alpha)^{2})}.\]  
Thus, the result, \(\square\)  
The next result gives a complementary lower bound for the generalized distance Estrada index \(D_{\alpha}E(G)\). The bound is characterized by the order \(n\) of graph, its Wiener index \(W(G)\) as well as the parameter \(\alpha\). Theorem 3: _Suppose that \(G\) is a connected graph of order \(n\). Then,_  
\[D_{\alpha}E(G)\geq e^{\frac{2(1-\alpha)W(G)}{n}}+(n-1)e^{\frac{-2(1-\alpha)W(G )}{n(n-1)}}, \tag{8}\]  
_where the equality holds if and only if \(G=K_{n}\)._  
**Proof. ** Thanks to Equation (1) and the well-known arithmetic-geometric mean inequality, we have  
\[D_{\alpha}E(G) = e^{\partial_{1}-\frac{2\alpha W(G)}{n}}+e^{\partial_{2}-\frac{2 \alpha W(G)}{n}}+\cdots+e^{\partial_{n}-\frac{2\alpha W(G)}{n}} \tag{9}\] \[\geq e^{\partial_{1}-\frac{2\alpha W(G)}{n}}+(n-1)\left(\prod_{i=2}^ {n}e^{\partial_{i}-\frac{2\alpha W(G)}{n}}\right)^{\frac{1}{n-1}}\] \[= e^{\partial_{1}-\frac{2\alpha W(G)}{n}}+(n-1)\left(e^{\frac{2 \alpha W(G)}{n}-\partial_{1}}\right)^{\frac{1}{n-1}}. \tag{10}\]Consider the following function,  
\[D_{\alpha}E^{2}(G)=\sum\limits_{i=1}^{n}e^{2\left(\partial_{i}-\frac{2\alpha W (G)}{n}\right)}+2\sum\limits_{i<j}e^{\left(\partial_{i}-\frac{2\alpha W(  
The function  
\[f(x)=\frac{1}{2}\left(x+\sqrt{(x-2n)^{2}+4(4-x)(1-\alpha)^{2}\sum_{1\leq i<j\leq n }(d_{ij})^{2}+2\alpha^{2}(4-x)P}\right),\]0  
for \(x\geq 0\).