The H - Line Signed Graph of a Signed Graph
نویسندگان
چکیده
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 connected graph H of order at least 3, the H-Line Graph of a graph G = (V,E), denoted by HL(G), is a graph with the vertex set E, the edge set of G where two vertices in HL(G) are adjacent if, and only if, the corresponding edges are adjacent in G and there exists a copy of H in G containing them. Analogously, for a connected graph H of order at lest 3, we define the H-Line signed graph HL(S) of a signed graph S = (G,σ) as a signed graph, HL(S) = (HL(G), σ), and for any edge e1e2 in HL(S), σ ′(e1e2) = σ(e1)σ(e2). In this paper, we characterize signed graphs S which are H-line signed graphs and study some properties of H-line graphs as well as H-line signed graphs.
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