Given a set S, and a family of nonempty subsets of S, the corresponding intersection graph has a vertex for each subset, and a connecting edge whenever two subsets intersect.
Any graph can be considered an intersection graph, by constructing S and its subsets appropriately. Create an element for each isolated vertex and each edge. Isolated points produce single element subsets. A vertex with positive degree defines a subset corresponding to its incident edges.
Since every graph is an intersection graph, the intersection number of a graph G is the size of the smallest set S, such that G is the intersection graph of S. Since intersection numbers can be computed per component, and added together, we will assume G is connected. The number of edges e is an upper bound, using the incident-edge subsets described above. However, the graph K16 has an intersection number of 5, far less than its 120 edges. Let each subset include the first element, and use the remaining four to establish 16 different subsets. Every pair of subsets intersects, and every pair of points is connected by an edge.
If G contains no triangles, then no three subsets can share a common element. Two adjacent vertices define the first set element, the element shared between them. The next adjacent vertex introduces another set element, and so on. The intersection number of a triangle free graph is e.