I'm reading about covering space ,i'm having the following two doubts:
1.Suppose $X$ be a graph and $\widetilde {X}$ be a regular cover/Normal Cover of $X$ other than $X$.Then show that $\widetilde{X}$ is a regular graph (i.e. each vertex has same degree).
2. Suppose if $X$ is finite then is it true that every covering graph of $X$ is regular graph?
I know that $\widetilde {X}$ is a graph but i can't see that its a regular graph.
"Regular" in the covering sense and "regular" in the graph sense are basically totally unrelated. Statement (1) is horribly false. Indeed, if $p:\widetilde{X}\to X$ is a covering of graphs, then for any vertex $v\in\widetilde{X}$, $v$ has the same degree as $p(v)$ (this is basically immediate from the fact that $p$ is a local homeomorphism, since the degree of a vertex is a local topological property). So, if you require your covering maps to be surjective, $\widetilde{X}$ is a regular graph iff $X$ is a regular graph (since their corresponding vertices have the same degrees).