The Adjacency Matrix of a Graph and its Powers
We define the adjacency matrix of a graph. Simply put if an edge connects vertex i to vertex j, then place a one in both the (i,j) and (j,i) entry of the matrix. If there is no edge between the vertices, then place a zero. This matrix of zeroes and ones while seemingly simple plays an essential role in important applications like the Google page rank algorithm. We show that powers of this matrix count the number of paths of different lengths from one vertex to another.
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