Based on this, we are thus only able to obtain the following sequences of vertices.
We now find the number of simple paths that are associated with the sequence (i) of the vertices as . In a similar way we obtain the number of simple paths that are associated to (ii) is, . The number of simple paths of the sequences associated to (iii) are given is,. Lastly the number of simple paths associated to (iv) is . Based on all these, we can thus obtain the number of simple path sequences that are connected to 1 and 5 to be given as. .
The graph above presents the tree diagram. Being a tree diagram each vertex given corresponds to a mentioned person (for instance, P stands for president, VP stands for vice-president, and so on). It is therefore clear that the many vertices represent the many persons mentioned. We see that for any two given vertices to be connected there must be a direct working relationship/association between the persons who corresponds to the given vertices.
For instance if by any chance the direct relationship/association between say the college president and the head of alumni relations is added, then we expect to obtain a graph which is not a tree, this is because, according to the definition, an undirected graph is called a tree if any of the two vertices are connected exactly by one simple path. In the obtained new graph however, the vertices AR and P are connected by two simple paths