# Edge Data Center Organization and Optimization by Using Cage Graphs
## 8 Conclusions  
In this study, a data center organization and optimization technique was developed for edge or fog environments, whose main feature is having two or three hops between a small number of physical hosts involved, as a given number of virtual hosts may be allocated to a physical host. This paper started with Introduction and Motivation sections, followed by an overview of graph theory, where we described the main types of graphs. After that, the concept of integrity of a graph was introduced as a measure of its reliability, which can be linked to the concept of network stability, where integrity values for some common type of graphs were provided. Afterward, cage graphs were presented as designs with the particularity that all nodes have the same values of regularity and girth, where the number of nodes for each available instance may have both a lower bound and an upper bound. In this sense, all known three-regular cage graphs were introduced, whose girth values ranged from 5 to 12. The most well-known \(r\)-regular cage graphs with \(r>3\) were also described, while their integrity values for topologies whose order grows up to 60 were listed. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Topology** & **Avg.** & **Mode** & **Median** & **Variance** & **Std.** & **Coefficient** \\  & **Hops** & & & & & **Dev.** & **of Variation** \\ \hline Full mesh & 1 & 1 & 1 & 0 & 0 & \(\infty\) \\
3-hypercube & 1.71 & 1.5 & 2 & 0.49 & 0.70 & 0.41 \\
4-hypercube & 2.13 & 2 & 2 & 0.78 & 0.88 & 0.41 \\ Folded 3-hyp. & 1.43 & 1 & 1 & 0.24 & 0.49 & 0.35 \\ Folded 4-hyp. & 1.67 & 2 & 2 & 0.22 & 0.47 & 0.28 \\
3-ary 2-cube & 1.50 & 1.50 & 1.50 & 0.25 & 0.50 & 0.33 \\
4-ary 2-cube & 2.13 & 2 & 2 & 0.60 & 0.77 & 0.36 \\ Petersen gr. & 1.67 & 2 & 2 & 0.22 & 0.47 & 0.28 \\ Heawood gr. & 2.07 & 2 & 2 & 0.53 & 0.73 & 0.35 \\ Robertson gr. & 1.89 & 2 & 2 & 0.32 & 0.57 & 0.30 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Statistics related to the average number of hops in each network topology. After that, the smallest cage graph (excluding complete graphs and complete bipartite graphs), namely the Petersen graph, also known as \((3,5)\)-cage graph,was proposed as a topology for data centers with 10 nodes, leading to a diameter of 2, which creates a really fast interconnection network among hosts. Hence, a scheme for linking together those 10 nodes was described, resulting in four case scenarios depending on the location of the source and destination nodes, which were denoted in both algorithmic and algebraic methods. Additionally, two other small cage graphs were studied: the Heawood graph, also known as the \((3,6)\)-cage graph, and the Robertson graph, also known as the \((4,5)\)-cage graph. The former is composed of 14 nodes, whereas the latter is composed of 19 nodes, although both have a diameter of 3. The former was described in an algorithmic and an algebraic manner, while only the algorithmic form of the latter was outlined. All three cage graphs proposed herein present a coefficient of variation among the lowest compared with those of other commonly used network topologies in data centers. In summary, the forwarding scheme proposed for a Petersen graph topology improves the performance compared with that of the traditional routing tables and MAC-address tables, as forwarding is achieved by integer divisions and modular arithmetic, as opposed to searching through the aforesaid tables for a matching entry, which requires more computation resources. Conceptualization, P.J.R. ; formal analysis, P.J.R. ; supervision, P.J.R., S.A., K.G., C.B. and C.J. ; validation, P.J.R. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: