line graph 
Let G be a simple connected graph with n vertices and m edges, LG be the line graph of G and λ1 (LG) ≥ λ2 (LG) ≥ … ≥ λm (LG) be the eigenvalues of the graph LG.


In this paper, the range of eigenvalues of a line graph is considered.


But by using the concept of edge cochromatic number it is proved that if G is the line graph of a connected trianglefree graph with ω(G)>amp;lt;5 and G≠K4, then z(G)≥χ(G)2.


It also allows for sensible definition of a line graph (the medial graph), particularly useful for domains consisting of connected channels, and not reliant on symmetries of the domains.


A special kind of embedding of graph into itsnth iterated line graph, called incidence embedding, is studied.


In this paper, we give a best possible Orelike condition for a graph so that its line graph is pancyclic or vertex pancyclic.


This paper shows that for a connected balanced digraph D and its line digraph L, if D is optimally super edgeconnected, then κ1(L) = 2λ1(D), and that for a connected graph G and its line graph L, if one of κ1(L) and λ2(G) exists, then κ1(L) = λ2(G)


Let K be the quasiLaplacian matrix of a graph G and B be the adjacency matrix of the line graph of G, respectively.


We study the cases, when a critical edge is removed from the line graph of a bipartite graph or from the complement of such a graph.


Online graph coloring of $${\mathbb{P}_5}$$free graphs


The concept of line perfection of a graph is defined so that a simple graph is line perfect if and only if its line graph is perfect in the usual sense.


In this paper we consider the problem of online graph coloring.


In an instance of online graph coloring, the nodes are presented one at a time.


The performance ratio of an online graph coloring algorithm for a class of graphsC is maxG ∈C(A(G)/χ(G)).


First Fit does as well as any online algorithm fordinductive graphs: we show that, for anyd and any online graph coloring algorithmA, there is adinductive graph that forcesA to use Ω(d logn) colors to colorG.


We also examine online graph coloring with lookahead.


A pseudoline graph(Γ, E) is a graph for which the vertices are the pseudolines of Γ and the edges are some unordered pairs of pseudolines of Γ .


We show that a graph is planar if and only if it is isomorphic to a diamondfree pseudoline graph.


The anticipation of the discrepancy at skeletal maturity is achieved with the Green and Anderson data, either with the arithmetic method, the growth remaining charts, or the Moseley straight line graph.


It is shown that every edgeclique graph is a clique graph, and that ifG is either an interval graph or a line graph, then so too isK(G).

