How does the binomial expansion helps understanding graph connectedness? I read this post about graph connectedness and adjacency matrix. But I do not understand how the last comment about using the binomial expansion would help. I know this approach is correct, but just do not fully grasp it.  
Would you mind to elaborate a little bit more? I did the binomial expansion and got $\binom{n-1}{0}A^{0}+\binom{n-1}{1}A^{1}+.....+\binom{n-1}{n-1}A^{n-1}$ which is a linear combination of all the As. but I still don't know how this helps. 

As far as I understand $$(I+A)^{n-1} = \sum_{k=0}^{n-1} \binom{n-1}k A^k$$
Only shows how to calculate $(I+A)^{n-1}$ and there is no some extra meaning in such expansion.
