## 6 Examples
Then,  
\[D_{\alpha}E(H)=n-1+e^{(1-\alpha)(5n-2r-4)}+e^{(\alpha-1)n}+\sum_{i=2}^{n}e^{( \alpha-1)(2\lambda_{i}+4)}.\]  
\(\Box\)  
The graph \(G\
abla G\) is obtained by joining each vertex of \(G\) to each vertex of a second copy of \(G\). **Corollary 5**. : _Let \(G\) be an \(r\)-regular graph with an adjacency matrix \(A\) and \(\mbox{spec}(G)=\{r,\lambda_{2},\ldots,\lambda_{n}\}\). Then, the generalized distance Estrada index of \(G\
abla G\) is_  
\[D_{\alpha}E(G\
abla G)=e^{(1-\alpha)(3n-r-2)}+e^{(1-\alpha)(n-r-2)}+2\sum_{i=2 }^{n}e^{(\alpha-1)(2\lambda_{i}+4)}.\]  
**Proof. ** For \(v\in G\
abla G\), it is easy to see that \(Tr(v)=d(v)+2(n-d(v)-1)+n=3n-d(v)-2=3n-r-2\). Then, \(G\
abla G\) is a transmission regular graph and \(Tr(G\
abla G)=(3n-r-2)I\). Note that the \(G\
abla G\) has distance spectrum (see in [37]) \(spec(G\
abla G)=\left\{3n-r-2,n-r-2,-2(\lambda_{i}+2)^{[2]}\right\}\), for \(i=2,\ldots,n\). Then,  
\[D_{\alpha}E(G\
abla G)=e^{(1-\alpha)(3n-r-2)}+e^{(1-\alpha)(n-r-2)}+2\sum_{i= 2}^{n}e^{(\alpha-1)(2\lambda_{i}+4)}.\]  
\(\square\)  
Next, we study the generalized distance Estrada index of the lexicographic product \(G[H]\) of two graphs \(G\) and \(H\). The lexicographic product of \(G\) and \(H\) can be defined as follows. **Definition 1** ([39]). : Let \(G\) and \(H\) be two graphs on vertex sets \(V(G)=\{u_{1},u_{2},\ldots,u_{p}\}\) and \(V(H)=\{v_{1},v_{2},\ldots,v_{n}\}\), respectively. Their _lexicographic product_\(G[H]\) is a graph defined by \(V(G[H])=V(G)\times V(H)\), the cartesian product of \(V(G)\), and \(V(H)\), where \(u=(u_{1},v_{1})\) is adjacent to \(v=(u_{2},v_{2})\) if and only if either  
(a) \(u_{1}\) is adjacent to \(v_{1}\) in \(G\), or  
(b) \(u_{1}=v_{1}\) and \(u_{2}\) is adjacent to \(v_{2}\) in \(G\). **Theorem 13**. : _Given a \(k\)-transmission regular graph \(G\) of order \(p\). If \(H\) is an \(r\)-regular graph of order \(n\) with adjacency eigenvalues \(\{r,\lambda_{2},\ldots,\lambda_{n}\}\), then_  
\[D_{\alpha}E(G)=e^{n(2n-r-2)-2\alpha(n^{2}-n-m)}+\sum_{i=2}^{n}e^{\alpha(n \lambda_{i}+rn+2n+2m)-n(\lambda_{i}+2)}.\]0  
_where \(\{\mu_{1},\ldots,\mu_{p}\}\) are the eigenvalues of the distance matrix \(D(G)\)._  
**Proof. ** For \(v\in G[H]\), it is easy to see that \(Tr(v)=r+2(n-r-1)+kn=kn+2n-r-2\). Then \(G[H]\) is a transmission regular graph and \(Tr(G[H])=(kn+2n-r-2)I\). Note that \(G[H]\) has distance spectrum (see [40]) \(spec(G[H])=\left\{n\mu_{i}+2n-r-2,-2(\lambda_{j}+2)^{[n]}\right\}\), for \(i=1,\ldots,p\) and \(j=2,\ldots,n\). Then,  
\[D_{\alpha}E(G)=e^{n(2n-r-2)-2\alpha(n^{2}-n-m)}+\sum_{i=2}^{n}e^{\alpha(n \lambda_{i}+rn+2n+2m)-n(\lambda_{i}+2)}.\]1  
\(\square\)  
**Example 1**. : _Let \(C_{n}\) be a cycle of order \(n\) and \(K_{2}\) be the complete graph of order 2. Then, the closed fence graph is defined as \(G=C_{n}[K_{2}]\), and depicted in \[D_{\alpha}E(G)=e^{n(2n-r-2)-2\alpha(n^{2}-n-m)}+\sum_{i=2}^{n}e^{\alpha(n \lambda_{i}+rn+2n+2m)-n(\lambda_{i}+2)}.\]2 Applying Theorem 13, we will be able to compute the generalized distance Estrada index of closed fence \(G=C_{n}[K_{2}]\). It is well known ([41], Theorem 3) that \(C_{n}\) is a \(k\)-transmission regular graph, with \(k=\lfloor\frac{n^{2}}{4}\rfloor\). It is also clear tht the adjacency spectrum of the graph \(K_{2}\) is \(spec(K_{2})=\{1,-1\}\).