# Edge Data Center Organization and Optimization by Using Cage Graphs
## 6 Applying Other Cage Graphs for Small Data Centers
### Heawood Graph  
Regarding the node distribution in a Heawood graph, also known as \((3,6)\)-cage graph, each node has just 3 links to its peers. Its overall shape is that of a tetradecagon, which is a polygon with 14 nodes and 14 edges; hence, each node has one link pointing out to its predecessor and another one to its successor. This context allows the sequential enumeration of all nodes from 0 to 13, such that if a node \(i\) is taken, then its predecessor link is to the node \((i-1)_{|14}\), whereas its successor link is to the node \((i+1)_{|14}\). Additionally, the third link alternatingly points to the node located five positions clockwise or counterclockwise in a sequential manner, which is why it is referred to as a remote port. Therefore, the expression for the third link is \((i+5\cdot(-1)^{i})_{|14}\), in a way that such a link in even nodes is forward, while in odd nodes, it is backward. Figure 5 depicts the node distribution of a Heawood graph, where all the aforesaid considerations are shown. Moreover, Figure 6 exhibits the destination of the three ports described in each node. Figure 5: Node distribution of a Heawood graph. An important feature of the Heawood graph is that the maximum distance between any pair of nodes is just three hops away. To start, the distance from a particular source node \(i\) to a given destination node \(j\) is measured and further adjusted to remain within the interval from \(-6\) to \(7\). Afterward, according to the adjusted distance and depending whether node \(i\) is even or odd, there are five scenarios to be studied on an individual basis, as shown in Algorithm 2. ```
\(dist\ =j-i\) if\(dist>7\)then \(dist=-(14-dist)\) elseif\(dist<-6\)then \(dist=-(-14-dist)\) endif if\(dist\in\{+1,+2,+3\}\)then  send data through switch \(i,port\ 1\) elseif\(dist\in\{-1,-2,-3\}\)then  send data through switch \(i,port\ 0\) elseif\(i\ mod\ 2=0\)then if\(dist\in\{+4,+5,+6,+7\}\)then  send data through switch \(i,port\ 2\) elseif\(dist\in\{-4,-5\}\)then  send data through switch \(i,port\ 1\) elseif\(dist=-6\)then  send data through switch \(i,port\ 0\) endif elseif\(i\ mod\ 2=1\)then if\(dist\in\{-4,-5,-6,+7\}\)then  send data through switch \(i,port\ 2\) elseif\(dist\in\{+4,+5\}\)then  send data through switch \(i,port\ 0\) elseif\(dist=+6\)then  send data through switch \(i,port\ 1\) endif endif
```  
**Algorithm 2**Moving from node \(i\) to node \(j\) in a Heawood graph. In order to obtain the formal algebraic model of a node in ACP, we must translate this algorithm into algebraic form by following the procedure shown for the ACP expression for the Petersen graph. In this way, (17) shows the formal algebraic model of a Heawood node \(V_{i}\), where an auxiliary function (16) is previously defined to maintain the value of the adjusted distance between source node \(i\) and destination node \(j\), whereas \(V_{i}\) uses that function in order to discriminate the five possible cases considered. Figure 6: Heawood graph node with ports going to their neighboring nodes. Given a node \(i\), one of the links must be with its predecessor, meaning node \((i-1)_{|19}\), and another one must be with its successor, meaning node \((i+1)_{|19}\). Regarding the remaining two links, one is with a remote node clockwise, whereas the other one moves to another remote node counterclockwise. Unfortunately, a regular pattern is not found to reach the destinations of both remote links no matter which node \(i\) is taken, although only some combinations are repeated in all cases, as shown in Table 6.  
Nodes 0 and 13 share the same movements for clockwise and counterclockwise links, being \(+4\) and \(-5\), respectively. In this sense, nodes 1 and 7 share the movements \(+8\) and \(-8\), whereas nodes 8 and 14 share \(+5\) and \(-4\). Nodes 3, 12, 16, and 18 share movements \(+8\) and \(-7\); nodes 2, 4, 6, 10, and 17 share \(+4\) and \(-4\), which are the most repeated.