A typical example of a directed graph is a social media site. The vertices of this graph

are all of the accounts, $a_i,$ and there is an edge between any two accounts $a_i$ and $a_j$ if

$a_i$ "follows" $a_j.$

Let $M$ denote the vertex matrix for this social media site:

$M_{ij}=\{\begin{array}{ccc}1& if{a}_i\text{f}\text{o}\text{l}\text{l}\text{o}\text{w}\text{s}& a_j\\ 0& \text{o}\text{t}\text{h}\text{e}\text{r}\text{w}\text{i}\text{s}\text{e}& \end{array}$

What is the interpretation of the $i,j$ entry of $M^{T}M?$ Explain carefully.

The six original NHL teams are, Detroit Red Wings, Toronto Maple Leafs, Montreal

Canadiens, Chicago Blackhawks, Boston Bruins and New York Rangers. We make a

dominance$-$directed graph as follows. The vertices are these $6$ teams, and we say that

Team ${?}_i$ dominates Team ${?}_j$ if the first has more victories over the second throughout the

history of their rivalry.

Rank these six teams in terms $($vertices$)$ in terms of the power of each vertex.