Why is this the obvious? (graph Theory)

[Firepanda]

http://www.win.tue.nl/~aeb/srgbk/node10.html

Under the 'For connected graphs all is clear from above'

I've been asked to prove this for connected graphs, but I don't see how this is so clear?

Can anyone help me? Thanks

 

[HallsofIvy]

It says "it is clear from the above" but you have cut off the "above"! What did it say before that theorem?

 

[Firepanda]

It says "it is clear from the above" but you have cut off the "above"! What did it say before that theorem?

Well I googled my problem and found that link after some pages. I'm not sure what it said.

But it probably was something like the Perron Frobenius theorem.

From my notes I have (Perron Frobenius Thereom)

'Let A be in the set of all Schrödinger operators, and suppose a graph G is connected.

Then eigenvalue(1) is simple (i.e. has multiplicity 1) and its eigenspace is generated by an eigenvector that is strictly positive everywhere.'My general question that I was asked to do came from this email he sent me:

'These graphs are bipartite, meaning that if you change the sign of every other entry in an eigenvector then you get an eigenvector associated with negative the first eigenvalue (prove this!)'