AHL 3.14—Graph theory

The sum of all vertex degrees equals $2|E|$.

$2+2+2+3+3+4+4=20$.

$|E|=20/2=10$.

The graph has $10$ edges.

$20$ is the sum of degrees, not divided by 2 - forgetting to halve.

$9$ would come from miscounting one degree value when summing.

$14$ does not correspond to any correct or half-correct calculation of this sum.

Only $10$ satisfies the Handshaking Lemma correctly.

In $K_n$, every vertex connects to every other vertex exactly once.

With $n=6$, each vertex connects to the remaining $6-1=5$ vertices.

Therefore $\mathrm{deg}(v)=5$ for every vertex.

The degree of any vertex in $K_6$ is $5$.

$6$ mistakenly counts the vertex itself as a neighbour, using $n$ instead of $n-1$.

$15$ is the total number of edges in $K_6$ ($\frac{6\times 5}{2}$), not a vertex degree.

$30$ is the sum of all degrees ($2\times 15$), not an individual degree.

Only $5$ correctly represents an individual vertex's degree.