Paths, Cycles And Independence Number Of Super Line Graphs

For any positive integer r, the super line graph L_r(G)of index r of a graph G is defined as follows: The vertices of L_r(G) are the r-subsets of E(G), and two distinct r-subsets S and T of E(G) are adjacent in L_r(G) whenever there exist adjacent edges s∈S and t∈T in G. Bagga, Beinake and Varma showed that if G is connected and has at least two edges, then L₂(G) is pancyclic. An improved result was also obtained that if G has no isolated edges, then L₂(G) is vertex-pancyclic. A connected graph G of order n is path-comprehensive if every pair of vertices are joined by paths of all lengths 2,3, ..., n - 1. In this paper, we obtain that if G has no isolated edges, then L₂(G) is path-comprehensive. Bagga, et al. asked whether L₂(G) is also vertex-pancyclic even if one isolated edge is permitted. We give a positive answer to the question by obtaining that if G has at most one isolated edge, then L₂(G) is vertex-pancyclic. At last, we discuss the independence number of super line graphs of arbitrary index.