In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this paper, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions a...

In 1982, Opsut showed that the competition number of a line graph is at most two and gave a necessary and sufficient condition for the competition number of a line graph being one. In this note, we generalize this result to the competition numbers of generalized line graphs, that is, we show that the competition number of a generalized line graph is at most two, and give necessary conditions an...

‎Let G=(V(G),E(G)) be a simple connected graph with vertex set V(G) and edge‎ ‎set E(G)‎. ‎The (first) edge-hyper Wiener index of the graph G is defined as‎: ‎$$WW_{e}(G)=sum_{{f,g}subseteq E(G)}(d_{e}(f,g|G)+d_{e}^{2}(f,g|G))=frac{1}{2}sum_{fin E(G)}(d_{e}(f|G)+d^{2}_{e}(f|G)),$$‎ ‎where de(f,g|G) denotes the distance between the edges f=xy and g=uv in E(G) and de(f|G)=∑g€(G)de(f,g|G). ‎In thi...

We present a new algorithm based on Dual Graph Contraction (DGC) to transform the Run Graph into its Minimum Line Property Preserving (MLPP) form which, when implemented in parallel, requires O(log(longestcurve)) steps. A MLPP graph of a line image compliments the structural information in geometric graph representations like the run graph. Using such a graph and its dual, line image analysis s...

We present a new algorithm based on Dual Graph Contrac tion DGC to transform the Run Graph into its Minimum Line Property Preserving MLPP form which when implemented in parallel requires O log longestcurve steps A MLPP graph of a line image compliments the structural information in geometric graph representations like the run graph Using such a graph and its dual line image analysis systems can...

In [Discrete Math. 230 (2001), 133-141], it is shown that Tutte’s 3-flow conjecture that every 4-edge-connected graph has a nowhere zero 3-flow is equivalent to that every 4-edge-connected line graph has a nowhere zero 3-flow. We prove that every line graph of a 4-edgeconnected graph is Z3-connected. In particular, every line graph of a 4-edge-connected graph has a nowhere zero 3-flow.

In recent articles by Grohe and Marx, the treewidth of the line graph of a complete graph is a critical example—in a certain sense, every graph with large treewidth “contains” L(Kn). However, the treewidth of L(Kn) was not determined exactly. We determine the exact treewidth of the line graph of a complete graph. C © 2014 Wiley Periodicals, Inc. J. Graph Theory 00: 1–7,

A Smarandachely k-signed graph (Smarandachely k-marked graph) is an ordered pair S = (G,σ) (S = (G,μ)) where G = (V, E) is a graph called underlying graph of S and σ : E → (e1, e2, ..., ek) (μ : V → (e1, e2, ..., ek)) is a function, where each ei ∈ {+,−}. Particularly, a Smarandachely 2-signed graph or Smarandachely 2-marked graph is called abbreviated a signed graph or a marked graph. Given a ...

let $r$ be a commutative ring with identity‎. ‎we use‎ ‎$varphi (r)$ to denote the comaximal ideal graph‎. ‎the vertices‎ ‎of $varphi (r)$ are proper ideals of r which are not contained‎ ‎in the jacobson radical of $r$‎, ‎and two vertices $i$ and $j$ are‎ ‎adjacent if and only if $i‎ + ‎j = r$‎. ‎in this paper we show some‎ ‎properties of this graph together with planarity of line graph‎ ‎assoc...