On Several Parameters of Super Line Graph <inline-formula><math display="inline"><semantics><mrow><msub><mi mathvariant="script">L</mi><mn>2</mn></msub><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>

Abstract

The super line graph of index r, denoted by Lr(G), is defined for any graph G with at least r edges. Its vertices are the sets of r edges of G, and two such sets are adjacent if an edge of one is adjacent to an edge of the other. In this paper, we give an explicit characterization for all graphs G with L2(G) being a complete graph. We present lower bounds for the clique number and chromatic number of L2(G) for several classes of graphs. In addition, bounds for the domination number of L2(G) are established in terms of the domination number of the line graph L(G) of a graph. A number of related problems on L2(G) are proposed for a further study.

Keywords

• super line graph
• clique
• coloring
• domination number