GE2-rings and a graph of unimodular rows

For a commutative ring A we consider a related graph, $\mathrm{\Gamma }(A)$, whose vertices are the unimodular rows of length 2 up to multiplication by units. We prove that $\mathrm{\Gamma }(A)$ is path-connected if and only if A is a ${\mathrm{GE}}_2$-ring, in the terminology of P. M. Cohn. Furthermore, if $Y(A)$ denotes the clique complex of $\mathrm{\Gamma }(A)$, we prove that $Y(A)$ is simply connected if and only if A is universal for ${\mathrm{GE}}_2$. More precisely, our main theorem is that for any commutative ring A the fundamental group of $Y(A)$ is isomorphic to the group $K_2(2,A)$ modulo the subgroup generated by symbols.