Relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$

Question

Let $E=E(G,S)$ be the graph defined by a group $G$ and a subset $S$ of $G$. What is relationship between group $C^\ast$-algebras $C^\ast(G)$ and graph $C^\ast$-algebras $C^\ast(E)$?