Finding binary operations on connected graphs

Question

If $G = (V,E)$ is a connected graph with $||V|| \geq 2$ , $W(G)$ being the set of all paths in $G$. How do you find a binary operation $ +$ on $W(G)$ such that $\langle W(G),+\rangle$ is an algebra and that there exists for all alphabets $||\sum|| \geq 1$ an injective homomorphism from $\langle W(G),+\rangle$ on $\langle \sum^*,\circ \rangle$?