Exercise 1, § 2

Let G be a graph and let $Q = (S,F, o, t)$ be its underlying quiver.

(a) Prove that one has $d_+(s) =d_-(s)$ for every $s\in S$. This integer is then called the degree of G at $s$ and denoted by $d(s)$.

(b) Suppose that G is finite. For there to exist a loop $(a_0, f_1, . . . , f_n, a_n)$ in G such that, for every edge $\varphi$ of G, there exists a unique integer $n$ such that $\varphi =\{f_n, f_n\}$, it is necessary and sufficient that G be connected and that $d(s)$ be pair, for every $s\in$ Som(G).