Let $\ (V\ E)\ $ be a graph, i.e. $\ E\subseteq\binom V2.\ $ A $2$-lift pattern of a graph is a function $\ e:E\rightarrow\{-1\,\ 1\}.\ $ The induced 2-lift is defined as the graph $\ V\times\{-1\,\ 1\}\,\ E\_e\ $ where


$$E\_e\:=\ \{\{(a\ s)\,\ (b\ t)\}\ :\ \{a\ b\}\in V\ \ and\ \ t=e(\{a\ b\})\cdot s\}$$ 


* Now by looking at the $2$-lift pattern can one say if the lifted graph is connected or not?



When is a $2$-lift of a graph connected?
Still, I don't understand the question. The matrix determines the graph, hence there is a way to tell whether it's connected. If the question is about **how** to do that, then here is the first thing that comes to my mind: 



> 
> the double is connected iff
>  (1) the original graph $G$ is connected, and (2) the homomorphism $H\_1(G)\to\Bbb Z\_2$ defined by the $2$-lift pattern is nontrivial.
> 
> 
> 


Condition (2) can be restated as follows: there is a $1$-cycle spanned by the original vertices $i\_1,i\_2,\ldots,i\_k,i\_{k+1}=i\_1$ such that $\prod\_{j=1}^k e(\{i\_j\ i\_{j+1}\})=-1$.


