Edge clique graphs and some classes of chordal graphs Original Research Article

The edge clique graph of a graph G is one having as vertices the edges of G, two vertices being adjacent if the corresponding edges of G belong to a common clique. We describe characterizations relative to edge clique graphs and some classes of chordal graphs, such as starlike, starlike-threshold, split and threshold graphs. In particular, a known necessary condition for a graph to be an edge clique graph is that the sizes of all maximal cliques and intersections of maximal cliques ought to be triangular numbers. We show that this condition is also sufficient for starlike-threshold graphs.