Graph Theory - Can two vertices have two distinct edges? My question is simple:
Can two vertices in a graph have two distinct edges connecting them? Why or why not?

 A: For a graph, or a simple graph, no, because a simple graph is a pair of sets, in which one is the set of vertices and the other is the set of edges. And elements of the set of edges are identified with endpoints, so, in this sense, there cannot be edges with same endpoints (if there are, then they are identified as the same one).
But there is also something called "multigraph," in which edges with the same endpoints are permitted. Check https://en.wikipedia.org/wiki/Multigraph. You may also want to look up "pseudograph."
A: I put this answer up only because OP asked for it in a comment, and also to present the other two terms I've heard of besides the widely used "simple graph".
In a graph, two or more edges connecting two distinct vertices are called parallel edges,
and an edge connecting only one vertex to itself is called a loop.
According to some sources: a graph allowing no loop or parallel edges is a "simple graph", a graph allowing parallel edges but no loops is a "multigraph", and if both parallel edges and loops are allowed the graph is a "pseudograph". [It's not clear to me whether a multigraph must have at least one case of a parallel edge, or whether a pseudograph which happens not to have loops would qualify as a multigraph.]
I also don't know whether there is general agreement about the definitions of multigraph and pseudograph. (Nor do I think it matters much provided a text/article makes the definitions used clear.)
A: It just depends on what you define a graph to be. Typically, graphs with at most one edge between vertices (and no loops) are referred to as simple graphs.
A: This structure is called a multigraph. Multigraphs are graphs except you can have an arbitrary number of edges between any pair of vertices and you can have edges where both ends are the same (this is called a loop).
