AbstractNot every edge-clique graph is a clique graph. 1 IntroductionLet Gbe a graph. The clique graph of G, K(G), is the intersection graph of thefamily of all maximal cliques of G. The edge-clique graph of G, K e (G), is theone whose vertices are the edges of G, two vertices being adjacent in K e (G),when the corresponding edges of G belong to a same clique. A graph G is aclique graph (edge-clique graph) if there exist a graph H such that K(H) = G(K e (H) = G).In a 1991 paper [4], Theorem 1, it is affirmed that every edge-clique graph isa clique graph (*). However, Prisner [1] noted that the proof of Theorem 1 isnot correct. In this note we shall prove that (*) does not hold, i.e. we showthat it is not always the case where an edge-clique graph is a clique graph.In Section 2 we review some properties of both clique and edge-clique graphsand observe that every edge-clique graph whose largest clique has size at mostthree is a clique graph. In Section 3 we show an edge-clique graph that is nota clique graph and prove that it is a minimum counterexample to (*).All graphs considered are finite, simple and undirected. The vertex and edgesets of a graph G are represented by V(G) and E(G), respectively. For C ⊆V(G), say that C is a clique when C induces a complete subgraph in G. Amaximal clique is one not properly contained in any other. Let ω(G) denote