## 4 Experimentation and Results
### Main Results of Cloud Computing Graph
#### 4.1.4 Theorem 2  
Let \(\mathrm{C_{rs}}=\mathrm{G}\) be a graph of the cloud computing network, then the CQI and QCI indices are  
\[\mathrm{CQI}(\mathrm{G})=\frac{\sqrt{\left(\left(s-1\right)^{2}\right)}\, \mathrm{r}\left(s-1\right)\left(s-2\right)}{2s-2}+\frac{\sqrt{2\left(\left(s- 1\right)^{2}+2\left(r+s-2\right)^{2}\right)}}{2s-3+\tau}\mathrm{r}\left(s-1 \right)+ \tag{9}\]  
\[\mathrm{QCI}(\mathrm{G})=\frac{\frac{1}{2}\left(2s-2\right)\left(r\left(s-1 \right)\left(s-2\right)\right)}{\sqrt{\left(s-1\right)^{2}}}+\frac{\left(2s- 3+r\right)\mathrm{r}\left(s-1\right)}{\sqrt{\left(2s-1\right)^{2}}+2\left(r+ s-2\right)^{2}}+\]  
\[\frac{1}{4}\frac{\left(2r+2s-4\right)\left(s-1\right)}{\sqrt{\left(r+s-2 \right)^{2}}} \tag{10}\]  
Equations (9) and (10) represent the proven results of the graph of the cloud computing network mentioned in Figure 2. These are the results that have been validated and are utilized for modeling the methods that reduce latency. The results are validated because Theorem 2 is proven, and the graphical results also show quite sharp upper and lower bounds of the graph of the cloud network by the CQI and QCI. So, these modeling outcomes exhibit the best qualities for the creation of improved cloud networks using the CQI invariants.  
\[\mathrm{CQI}(\mathrm{G})=\sum_{\mathrm{uv}\in\mathrm{E}(\mathrm{G})}\frac{ \sqrt{2\left(\mathrm{d_{G}}(\mathrm{u})^{2}+\mathrm{d_{G}}(\mathrm{v})^{2} \right)}}{\mathrm{d_{G}}(\mathrm{u})+\mathrm{d_{G}}(\mathrm{v})}\]  
\[\mathrm{CQI}(\mathrm{G})=\frac{\sqrt{2\left(\left(s-1\right)^{2}+\left(s-1 \right)^{2}\right)}\frac{\left(r\left(s-1\right)\left(s-2\right)\right)}{s-1+ s-1}}{\frac{\sqrt{2\left(\left(s-1\right)^{2}+\left(r+s-2\right)^{2}\right)}}{s-1+s-1+s-2}} \mathrm{r}\left(s-1\right)\]  
\[+\frac{\sqrt{2\left(\left(r+s-2\right)^{2}+\left(r+s-2\right)^{2}\right)}}{r \frac{s-2\left(r+s-2\right)^{2}}{r+s-2+r+s-2}}\frac{\left(r\left(s-1\right) \right)}{2}\]  
Figure 4: Results of KBSO and KBSO\({}_{\mathrm{red}}\) invariants for cloud computing network.  
\[\text{CQI}(\text{G})=\frac{\sqrt{\left((s-1)^{2}\right)}\text{r}(s-1)(s-2)}{2s-2}+ \frac{\sqrt{2\left((s-1)^{2}+2(r+s-2)^{2}\right)}}{2s-3+r}\text{r}(s-1)\]  
\[\text{QCI}\left(\text{G}\right)=\sum_{\text{uv}\in\text{E}(\text{G})}\frac{ \left(\text{d}_{\text{G}}(\text{u})+\text{d}_{\text{G}}(\text{v})\right)}{ \sqrt{2\left(\text{d}_{\text{G}}(\text{u})^{2}+\text{d}_{\text{G}}(\text{v})^ {2}\right)}}\]  
\[\text{QCI}(\text{G})=\frac{\text{s}-1+s-1}{\sqrt{2\left((s-1)^{2}+(s-1)^{2} \right)}}\frac{\left(\text{r}(s-1)(s-2)\right)}{2}\]  
\[+\frac{\text{s}-1+r+s-2}{\sqrt{2\left((s-1)^{2}+(r+s-2)^{2}\right)}}\text{r}( s-1)\]  
\[+\frac{\text{r}+s-2+r+s-2}{\sqrt{2\left((r+s-2)^{2}+(r+s-2)^{2}\right)}}\frac {\left(\text{r}(s-1)\right)}{2}\]  
\[\text{QCI}(\text{G})=\frac{\frac{1}{2}\left(2s-2\right)r(s-1)(s-2)}{\sqrt{(s -1)^{2}}}+\frac{\left(2s-3+r\right)r(s-1)}{\sqrt{2(s-1)^{2}+2(r+s-2)^{2}}}\]  
\[+\frac{1}{4}\frac{\left(2r+2s-4\right)r(s-1)}{\sqrt{(r+s-2)^{2}}}\]  
Figure 5 shows the results with clear upper and lower bounds Equations (9) and (10) of the CQI and QCI in red and blue colors, respectively, in the 3D version. As seen in Figure 5, the separation and straight line started from one to three for the 'r' and's' parameters. This would be used accordingly during the construction of cloud computing networks.