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

For a connected graph H of order at least 3, the H-line graph of a graph is defined as that graph whose vertices are the edges of G and where two vertices are adjacent if and only if the corresponding edges of G are adjacent and belong to a common copy of H . In particular, when H = P3, the H-line graph HL(G) is the standard line graph L(G) and for any connected graph G on at least three vertic...

The critical group of a graph is a finite abelian group whose order is the number of spanning forests of the graph. This paper provides three basic structural results on the critical group of a line graph. • The first deals with connected graphs containing no cut-edge. Here the number of independent cycles in the graph, which is known to bound the number of generators for the critical group of ...

can be interpreted as matching of an input graph (keyword) with a large set of graphs (document). More formally, in order to spot a certain keyword wi, all p graph instances gi1, ... , gip of that word wi occurring in the training set are matched against all graph words in each text line using our adapted graph matching procedure. That is, for a given word wi and a specific text line s pairwise...

Thomassen conjectured that every 4-connected line graph is Hamiltonian. A vertex cut X of G is essential if G−X has at least two non-trivial components. We prove that every 3-connected, essentially 11-connected line graph is Hamiltonian. Using Ryjác̆ek’s line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is Hamiltonian. © 2005 Elsevier Inc. All rights...

It is shown that the existence of a Hamilton cycle in the line graph of a graph G can be ensured by imposing certain restrictions on certain induced subgraphs of G. Thereby a number of known results on hamiltonian line graphs are improved, including the earliest results in terms of vertex degrees. One particular consequence is that every graph of diameter 2 and order at least 4 has a hamiltonia...