# Edge Data Center Organization and Optimization by Using Cage Graphs
## 4 Advantages of a Petersen Graph to Boost Performance in Small Data Centers  
Using a cage graph as a topology for a data center may bring several advantages, such as the aforementioned integrity values, making the underlying network more resilient and sustainable, as well as having steady values of regularity and girth, thus making the length of the available paths between any particular pair of nodes more predictable. However, the most interesting feature is the short values of network diameter, which is the maximum distance between any two nodes.  
As edge computing admits a data center layout much simpler than their counterparts in the cloud, and even in the fog, because of the lower number of users involved within edge domains, the most convenient layout for edge environments using cage graphs seems to be the Petersen one. The Peterson is the smallest cage graph, allowing all users may be dealt with within just 10 nodes. Complete graphs and complete bipartite graphs are omitted so as to avoid full mesh-like topologies and to try and simplify the data center designs as much as possible.  
Hence, as stated above, the order of a \((3,5)\)-cage graph is 10, which is the number of nodes involved in the topology, and the integrity is 6. Moreover, every node is three-regular, as each of those vertices have three links toward their counterparts, whereas the girth is five, such that the shortest cycle between any couple of vertices is five. The diameter is just two, resulting in only two hops away between a given source node and a particular destination node.  
In this way, if an edge data center is composed of 10 nodes being interconnected as a Petersen graph, such a design may take advantage of an integrity value of six, which  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Order** & **Symbol** & **Nonisomorphic** & **Integrity** \\  & & **Instances** & \\ \hline
10 & \((3,5)\) & 1 & 6 \\
14 & \((3,6)\) & 1 & 8 \\
19 & \((4,5)\) & 1 & 11 \\
24 & \((3,7)\) & 1 & 12 \\
26 & \((4,6)\) & 1 & 14 \\
30 & \((3,8)\) & 1 & 14 \\
30 & \((5,5)\) & 4 & 18 or 19 \\
40 & \((6,5)\) & 1 & 25 \\
42 & \((5,6)\) & 1 & 22 \\
50 & \((7,5)\) & 1 & 30 \\
58 & \((3,9)\) & 18 & 24 \\
70 & \((3,10)\) & 3 & 28 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Integrity of small cage graphs.  
makes for a really resilient layout, as diverse links may be down while not affecting the overall performance must, and a diameter value of two, which brings any node destination within only a two-hop upper bound.  
Focusing on edge data centers, the aforesaid nodes may be considered as physical nodes, each of those keeping virtual nodes within. The latter may be migrated toward another physical node in order to stay as close as possible to their associated users in a moving IoT environment, which may be achieved in one or two hops, thus resulting in a network fast to be traversed and reducing the migration times, which also implies a higher energy saving rate.  
The cage graphs are not planar graphs by definition, as the latter may be represented in a bidimensional layout without any pair of crossing links, which is not possible when dealing with the former. Hence, a Petersen graph may be drawn in a couple of alternative ways, such as in Figure 1, where a combination of a pentagon and a pentagonal star is exhibited, or in Figure 2, where a hierarchical layout is depicted, thus relieving the diameter of two for any pair of nodes, even though the first one is preferred for clarity purposes when assigning node identifiers so as to describe the forwarding strategy among nodes.