## 4 Experimentation and Results  
The 'C\({}_{\text{r}\text{s}}\)' cloud network was mapped, converted into the graph, solved, validated, and the results optimized according to the steps and methodology discussed in the methodology section, with the help of topological invariants mentioned in Equations (1)-(6).  
\[\text{KBSO}(\text{G})=\sum_{\text{ue}}\ \sqrt{\text{d}_{\text{u}}^{2}+\text{d}_{ \text{p}}{}^{2}} \tag{1}\]  
\[\text{KBSO}_{\text{r}\text{p}}\underset{-}{\text{d}}\ (\text{G})=\sum_{\text{ue}}\ \sqrt{\left(\text{d}_{\text{u}}-1\right)^{2}+\left(\text{d}_{\text{p}}-1 \right)^{2}} \tag{2}\]  
Equations (1) and (2) show the KBSO index and its reduced form, which will be used for the solution of the cloud computing network. In the above equations, \(\text{d}_{\text{u}}\) and \(\text{d}_{\text{e}}\) are showing edge partitions, where 'u' and 'e' are the vertices of the graph \(\text{C}_{\text{r}\text{s}}\) under discussion.  
\[\text{CQI}(\text{G})=\sum_{\text{uv}\in\text{E}(\text{G})}\ \frac{\sqrt{2\left(\text{d}_{ \text{G}}(\text{u})^{2}+\text{d}_{\text{G}}(\text{v})^{2}\right)}}{\text{d}_{ \text{G}}(\text{u})+\text{d}_{\text{G}}(\text{v})} \tag{3}\]  
\[\text{QCI}(\text{G})=\sum_{\text{uv}\in\text{E}(\text{G})}\cdot\frac{\left( \text{d}_{\text{G}}(\text{u})+\text{d}_{\text{G}}(\text{v})\right)}{\sqrt{2 \left(\text{d}_{\text{G}}(\text{u})^{2}+\text{d}_{\text{G}}(\text{v})^{2} \right)}} \tag{4}\]  
Equations (3) and (4) show the CQI and QCI, which will be used for the solution of the cloud computing network. In the above equations, \(\text{d}_{\text{u}}\) and \(\text{d}_{\text{v}}\) are showing the edge partitions, where the vertices of the graph \(\text{C}_{\text{r}\text{s}}\) are the 'u' and 'v' under discussion.  
\[\text{D}(\text{G})=\sum_{\text{ue}}\ \sqrt{\text{d}\text{u}^{3}+\text{d}\text{v}^{ 3}} \tag{5}\]  
The cloud computing network is converted into graphical form first and then associated with the graph. The graph is solved through the KBSO index, CQI, Dharwad index, and their other forms.  
\[\text{RD}\ (\text{G})=\sum_{\text{ue}}\ \sqrt{\left(\text{d}\text{u}-1\right)^{ 3}+\left(\text{d}\text{v}-1\right)^{3}} \tag{6}\]  
Equations (5) and (6) show the Dharwad index and its reduced form will also be used for the solution of the cloud computing network. In the above equations, \(\text{d}_{\text{u}}\) and \(\text{d}_{\text{v}}\) are showing the edge partitions where the vertices of the graph \(\text{C}_{\text{r}\text{s}}\) are the 'u' and 'v' under discussion.  
Table 1 describes the edge partitions of the graph of the cloud computing network \(\text{C}_{\text{r}\text{s}}\), as given in Figure 1. Where 'e' represents the edge, the \(\text{d}_{\text{u}}\) and \(\text{d}_{\text{v}}\) are showing the edge partitions, and 'u' and 'v' are the vertices of the graph \(\text{C}_{\text{r}\text{s}}\) under discussion. Recurrence means the number of edges attached to a particular vertex, also called frequency. By dividing a graph's collection of nodes into mutually exclusive groups, a cloud graph partition reduces the network to a smaller graph. The partitioned graph will have edges made up of the original graph edges that cross over into the groups. The partitioned graph may be more useful for analysis and problem-solving than the original if the total number of edges is lower than in the case of the original graph.