Practical importance of interval / circular-arc graphs in resource allocation
Making Louis' comment into an answer:

Given a set of intervals (interval system) $\mathcal{I} = \{ [x_1, x_2] ,\dots, [x_{2n-1}, x_{2n}] \}$. Each of the $n$ intervals corresponds to the time period a certain job is being worked on. Now, we want to find all time periods in which the number of concurrently active jobs is maximal. These time periods happen to correspond to the maxcliques of maximal size of the intersection graph $G(\mathcal{I}) = (\mathcal{I} , E)$ with $E$ being defined as
$$ ([a,b] , [c,d]) \in E \iff [a,b] \cap [c,d] \neq \emptyset $$
The closed neighborhood of a vertice $v$ in a graph $G$ is defined as 
$$ N[v] := \{ u \in V(G) : \{v,u\} \in E \} \cup \{v\}$$
It is known that for any interval graph every maxclique can be characterized as the intersection of two neighborhoods, i.e. for each maxclique $C$ of $G$ there exist some $u,v \in V(G)$ s.t. $C = N[u] \cap N[v]$.

With this we can easily devise an efficient algorithm to find the wanted time periods.

If you have any other answers along these lines I'd be happy to hear them.