Topological properties of the representation may be reflected in the intersection graph. Even metric properties, like distance, sometimes show through.
Why is every power of an interval graphs an interval graphs, but not every power of a unit interval graph a unit interval graph? Why are odd powers of chordal graphs chordal, but the even powers not necessarily?
Let G be the intersection graph of the hypergraphs H=(A,(S_{x}/x V)). Then distances in G can be expressed by set intersection of the sets S_{x}^{t}, where S_{x}^{1} :=S_{x} and S_{x}^{t} is recursively defined. We get d_{G}(x,y) i+j1 if and only if S_{x}^{i} S_{y}^{j} is nonempty. Therefore (A,(S_{x}^{t}/x V)) is a intersection representation of the (2t1)th power of G, whose vertices are adjacent if and only if d_{G}(x,y) 2t. In some situations, these sets S_{x}^{t} have the same shape than the original sets S_{x}. For instance, if the only property P the sets S_{x} have to be obey is, that they should be connected subsets of some topological space, then the sets S_{x}^{t} are also connected subsets of it. Then every odd power of a graph in G(P) lies again in G(P). A prominent example is the classe of chordal graphs
[BP83], [D84]: Odd powers of chordal graphs are chordal. 
But we get an odd power result also for circulararc graphs or interval graphs in this way.
What about even powers? Here we need geometric models with ordering. Assume G is the intersection graph of Psets S_{x}, where for every two disjoint sets, one is to the left of the other. Assume furthermore that S_{y} \ S_{x} divides naturally into two parts, one to the left and the other to the right of S_{x}. Then we may define R_{x} as the union of S_{x} and all right parts of S_{y} \ S_{x} for neighbors y of x. It turns out that these sets R_{x} are an intersection representation of G^{2}. The only problem may be that these sets are not necessarily Psets. In some cases, however, we may replace them by Psets having the same intersection pattern than the sets R_{x}.
If the S_{x}^{t}construction above applies also, we get representations of all powers of G by Psets in that case.
This approach works for intervals on the real line without further ado. It works for trapezoids between two parallel lines by taking the convex hulls of the sets R_{x} constructed (which are no trapezoids). It even works for structures with some cyclic ordering, as for connected subsets (circular arcs) of the unit circle.

A. Raychaudhouri posed the problem whether or not it is true that whenever G^{k} is a circulararc graph, then G^{k+1} must too be a circulararc graph. 
There are several other situations where the distances transfer, let me just mention one more example for line graphs and clique graphs:
[H84], [H86]: For every graph G, diam(G)1 diam(L(G) diam(G)+1, as well as diam(G)1 diam(C(G) diam(G)+1. 
Back to the start page for intersection graphs.