# Why is the claw=$K_{1,3}$ not a line graph?

I understand the definition of a line graph as: The line graph of a graph $G$ has vertex set $E(G)$ with two vertices being adjacent if and only if the corresponding edges of $G$ share a vertex. In a claw there are three edges and 4 vertices and in the line graph there are still 3 edges but only 3 vertices, is this why it is not a line graph?