A digraph G is called a dominance-directed graph if for any pair of distinct vertices u and v of G, either u→v or v→u, but not both (here the notation u→v means there is an edge from u to v). Below is an example of a dominance-directed graph. A, B, C, D and E are five sports teams. Teams play each other exactly once, with no ties allowed. A -> B means team A beats team B.

In a dominance-directed graph, we define the power of a vertex as being the total number of 1-step and 2-step connections to other vertices. For example, the power of A is 4. Using the adjacency matrix and its square, calculate the power of each vertex and rank each team according to their vertex power.(show your work)

 

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A E Graph H D A E Graph H D