The structure of super line graphs

For a given graph G = (V, E) and a positive integer k, the super line graph of index k of G is the graph S/sub k/(G) which has for vertices all the k-subsets of E(G), and two vertices S and T are adjacent whenever there exist sϵS and tϵT such that s and t share a common vertex. In the super line multigraph L/sub k/(G) we have an adjacency for each such occurrence. We give a formula to find the adjacency matrix of L/sub k/(G). If G is a regular graph, we calculate all the eigenvalues of L/sub k/(G) and their multiplicities. From those results we give an upper bound on the number of isolated vertices.