Covering space of a graph is again a graph - why??

Question

i want to prove the statement in the heading. Thus given graph $G$ and a covering space $p:\tilde{X}\rightarrow G$ to prove that $\tilde{X}$ is also a graph. My idea was to take $\tilde{E}=p^{-1}(E)$ and $\tilde{V}=p^{-1}(V)$ with $E$ and $V$ the sets of edges and vertices resp. But how to go further? Have i have to use the lifting properties??`Can someone give me help with this question?! Thanks