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**HallsofIvy** Have you found A^2? That is, have you multiplied the matrix A by itself?

One thing I notice immediately is that "first row times first column" is 1+1= 2 so the "11" component of A^2 is 2. I don't know how you have labled the vertices in your graph but if we label them "1", "2", etc. The fact that the first row is "0 1 1 0 0 0" means there is a "bridge", line, from 1 to 2 and from 1 to 3 but no others. Of course that means that there is also a bridge from 2 to 1 and from 3 to 1 so those same numbers appear in the first column. I could go from 1 to 2 and then back again or I could go from 1 to 3 and then back again: there are 2 paths of length 2 from 1 to itself. Now, where should you look to find how many paths of length 2 from 1 to 3?