  Let $G$ be a finite simple graph on $n$ vertices, that contains no isolated
vertices, and let $I(G) \subseteq S = K[x_1, \dots, x_n]$ be its edge ideal. In
this paper, we study the pair of integers that measure the projective dimension
and the regularity of $S/I(G)$. We show that if the projective dimension of
$S/I(G)$ attains its minimum value $2\sqrt{n}-2$ then, with only one exception,
the its regularity must be 1. We also provide a full description for the
spectrum of the projective dimension of $S/I(G)$ when the regularity attains
its minimum value 1.
