One vertices and one edges in graph theory

Question

Is a figure with only one vertices and one edge connected to that vertices known as a graph ?

Answer 1

That is not a graph. A simple graph with one edge need to have at least 2 vertices.

Perhaps you meant this:

The graph $K_n$-clique is a graph in which all vertices are connected. Perhaps you want $K_2$.

Answer 2

If that edge is a loop, yes. It is a Multigraph. Otherwise, it is not a graph.

Answer 3

With the strict definition of a simple graph, then there is no graph with one vertex and one edge. However, you could define a generalization of simple graphs which allows "half" edges with only one vertex and is not a loop. As far as the combinatorics, a half edge is no different than a loop, except in a path, it would be a dead end. However, in a drawing of the generalized graph on a surface, the loop would be closed curve, while a half edge would be be an open curve with only one of its endpoints to be considered a vertex. How useful this would be is not obvious to me now, but there is no reason why it could not have some useful purpose, just as graphs themselves have developed a great number of uses.