## Where $$x$$ and $$y$$ are non-adjacent pairs in a simple connected graph prove that $$deg(x) + deg(y) \geq n-1$$

Am i reading something wrong? This was a workshop question given to me in preparation for exams but I don’t see how this is true.

A simple connected graph is a graph such that given any two points on the graph $$a$$, $$b$$ we can find a path from $$a$$ to $$b$$, without multiple edges and no loops. Take a graph $$n=4$$ for example:

If i let $$b$$ and $$d$$ be my $$x$$ and $$y$$ then the equation holds as $$deb(b) + deg(d) = 1 + 2 = 3 \geq 4-1 = 3$$

However, if we remove the edge between $$a$$ and $$d$$ then is the graph not still a simple connected graph where $$d$$ and $$b$$ are non-adjacent? Because in this case $$deg(b) + deg(d) = 2$$ which is less than $$3$$. Therefore the equation doesn’t seem to hold. I’m not quite sure what I’m missing here.