The structure of super line graphs

Abstract

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

On Several Parameters of Super Line Graph L2(G)

“…In 2008, Li, Li, and Zhang [15] showed that if G has no isolated edges, then L 2 (G) is path-comprehensive, and that if G has at most one isolated edge, then L 2 (G) is vertex-pancyclic, answering a question posed by Bagga, Beineke, and Varma [14]. We refer to [16][17][18][19][20][21][22][23][24][25][26] for more results on super line graphs.…”