A $2-$ lift of a graph is specified by a $\pm 1$ assignment on the edges of the graph ( given as a $\vert V\vert \times \vert V\vert$ signing matrix $A_s$) denoting which edge is to be duplicated by the identity permutation on two elements and which is to be lifted with a flip. Like if the $(a,b)$ entry of $A_s$ is $-1$ then it means that the edge $(a,b)$ will be replaced by $4$ vertices $a_1,a_2,b_1,b_2$ and edges $(a_1,b_2)$ and $(a_2,b_1)$. But if that entry is $1$ then the new edges would be $(a_1,b_1)$ and $(a_2,b_2)$.

• Now by looking at the signing matrix can one say if the lifted graph is connected or not?