TITLE: General recipe for building C*-algebras out of combinatorial object
QUESTION [9 upvotes]: I want to ask what should be a nice way to build C*-algebras out of objects like groups, inverse-semigroups, semigroups, ringgs or graphs. I know there are well known construction of C*-algebras out of those objects; but I want to understand what philosophy lies under the recipe. 
Say for discrete groups I know group C*-algebras are made of as universal object of unitary elements of groups elements with relations coming from group operation. Now what I understand is they take unitary because if we fix an element $g$ in the group $G$, the map $T_g:G\rightarrow G$ defined by $T_g(h)=gh$ is one-one onto morphism. For Ring C*-algebras also this works. In that case where the map is not onty they take isometry. But my confusion starts with graph C*-algebra. I don't understand why they take projections for vertices and partial isometrys for edges (and the given relations).

REPLY [6 votes]: Benjamin's answer nicely describes where the relations for graph $C^\ast$-algebras come from.
Here is an attempt to answer the question suggested by the title: what is the general recipe for constructing $C^\ast$-algebras from other mathematical objects.  This may not work for every type of $C^\ast$-construction, but it seems to be the general formula behind at least group $C^\ast$-algebras, crossed products, graph $C^\ast$-algebras (and more generally, $C^\ast$-algebras of inverse semigroups), semigroup $C^\ast$-algebras, and ring $C^\ast$-algebras.
To the mathematical object in question is associated a natural and concrete Hilbert space and a natural collection of operators on that Hilbert space (often called the regular representation).  Take the obvious $*$-algebraic relations that hold between these operators; then these are the relations used to define the universal $C^\ast$-algebra.
Here is this formula worked out for a few examples:
Group $C^\ast$-algebras: For a discrete group $G$, the natural concrete Hilbert space is $\ell^2(G)$ (with canonical ONB $(\xi_g)_{g \in G}$), and to each group element $g \in G$ is associated the operator $\lambda_g$ defined by
$$ \lambda_g(\xi_h) = \xi_{gh}. $$
The obvious relations here are, that $\lambda_g$ is a unitary and $\lambda_g\lambda_h = \lambda_{gh}$; that is to say, that $\lambda$ is a group homomorphism between $G$ and the unitary group of $\mathcal{B}(\ell^2(G))$.
Of course, the $C^*$-algebra generated by $\{\lambda_g\}$ is called the reduced $C^\ast$-algebra of $G$, while the universal one (generated by $\{u_g: g \in G\}$ satisfying the relations that they are unitary and $u_gu_h = u_{gh}$) is called the (full) group $C^*$-algebra of $G$.
Dynamical systems: Though this can be done more generally, let's stick to a discrete group $G$ acting by homeomorphisms $\alpha_g$ on a compact metric space $X$.  The natural concrete Hilbert space is $\ell^2(G \times X)$ (with ONB $(\xi_{g,x})_{g \in G, x \in X}$).  For each $g \in G$, we may associate the operator $\lambda_g$ defined by
$$ \lambda_g(\xi_{h,x}) = \xi_{gh,x}. $$
For each $f \in C(X)$, we may also associate the operator $D_f$ defined by
$$ D_f(\xi_{g,x}) = f(g^{-1}x)\xi_{g,x}. $$
The obvious relations are: that $g \mapsto \lambda_g$ is a group homomorphism from $G$ to the unitary group; that $f \mapsto D_f$ is a $*$-homomorphism; and that
$$ \lambda_g D_f \lambda_g^* = f \circ \alpha_g. $$
Graph $C^\ast$-algebras: The details of this construction are more or less explained in Benjamin's answer.  The concrete Hilbert space has an orthonormal basis $(\xi_{\pi})$ indexed by the nonempty paths in the graph.  An edge $e$ gives rise to the operator $v_e$ given by
$$ v_e(\xi_{\pi}) = \begin{cases} \xi_{e\pi},\ &\text{if $e\pi$ is a path,} \\
 0,\ &\text{otherwise.} \end{cases} $$
A vertex $v$ gives rise to the operator $p_v$ which is the projection onto the span of 
$$ \{\xi_{\pi}: \pi \text{ is a non-empty path beginning at $v$}\}. $$