We consider a generalization of edge intersection graphs of paths in a tree. Let P be a collection of nontrivial simple paths in a tree T . We define the k -edge ( k ⩾ 1 ) intersection graph Γ k ( P ) , whose vertices correspond to the members of P , and two vertices are joined by an edge if the corresponding members of P share k edges in T . An undirected graph G is called a k -edge intersection graph of paths in a tree, and denoted by k -EPT, if G = Γ k ( P ) for some P and T . It is known that the recognition and the coloring of the 1-EPT graphs are NP-complete. We extend this result and prove that the recognition and the coloring of the k -EPT graphs are NP-complete for any fixed k ⩾ 1 . We show that the problem of finding the largest clique on k -EPT graphs is polynomial, as was the case for 1-EPT graphs, and determine that there are at most O ( n 3 ) maximal cliques in a k -EPT graph on n vertices. We prove that the family of 1-EPT graphs is contained in, but is not equal to, the family of k -EPT graphs for any fixed k ⩾ 2 . We also investigate the hierarchical relationships between related classes of graphs, and present an infinite family of graphs that are not k -EPT graphs for every k ⩾ 2 . The edge intersection graphs are used in network applications. Scheduling undirected calls in a tree is equivalent to coloring an edge intersection graph of paths in a tree. Also assigning wavelengths to virtual connections in an optical network is equivalent to coloring an edge intersection graph of paths in a tree.