Is G necessarily a cycle?
I suspect not but I'm having hard time showing this.
Also,
Let be a tree. Prove that the average degree of a vertex in T is less than 2.
I know that the sum of degrees of all vertices is $2|E|=2|V|-2$. Thus the graph must be connected, and the average degrees of a vertex is less than 2, so the vertices must be a degree of one. Is this correct?
Best Answer
1) No. Just take two triangles.
If $G$ is connected and every vertex has degree $2$ then $G$ is a cycle.
2) Combine sum of degrees formula with tree formula...