Graphs that every two edges have a common vertex

Question

Do the following cases enumerate all possible cases when an undirected graph that every two edges have common vertex?

1. an undirected graph with three vertices, which is also a clique.

2. an undirected graph with each edge connecting each vertex to a particular vertex, forming a star.

Thanks.

Answer 1

Suppose that there are more than 1 vertex, $v_1, v_2$, st $deg\, ( v_1), deg \, (v_2) \geq 2 $. Conclude they are connected, and show that there is some triangle imbeded in the graph. Then conclude that any other edge sits on two of the three vertexes of the triangle.

This always assuming that there is some edge in each vertex, and there are no parallel edges.

Answer 2

There could be isolated vertices.