Size of Star Graph

Theorem

Let $S_n$ denote the star graph of order $n$ where $n > 0$.

The size of $S_n$ is given by:

$\size {S_n} = n - 1$

Proof

By definition of star graph, one vertex is adjacent to all the $n - 1$ other vertices.

So the size of $S_n$ is at least $n - 1$.

All the other vertices are of degree $1$, and the edges they are incident with are the ones joining them to the distinguished vertex.

Those edges have already been counted.

So the size of $S_n$ is exactly $n - 1$.

$\blacksquare$