Geodesic
On •-Line Signed Graphs L•(S)
Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 215-227

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A signed graph (or sigraph for short) is an ordered pair S = (Su,σ), where Su is a graph, G = (V,E), called the underlying graph of S and σ : E → +,− is a function from the edge set E of Su into the set +,−. For a sigraph S its •-line sigraph, L•(S) is the sigraph in which the edges of S are represented as vertices, two of these vertices are defined adjacent whenever the corresponding edges in S have a vertex in common, any such L-edge ee′ has the sign given by the product of the signs of the edges incident with the vertex in e ∩ e′. In this paper we establish a structural characterization of •-line sigraphs, extending a well known characterization of line graphs due to Harary. Further we study several standard properties of •-line sigraphs, such as the balanced •-line sigraphs, sign-compatible •-line sigraphs and C-sign-compatible •-line sigraphs.
Keywords: sigraph, line graph, •-line sigraph, balance, sign-compatibility, C-sign-compatibility 
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     author = {Sinha, Deepa and Dhama, Ayushi},
     title = {On {{\textbullet}-Line} {Signed} {Graphs} {L\protect\textsubscript{{\textbullet}}(S)}},
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Sinha, Deepa; Dhama, Ayushi. On •-Line Signed Graphs L•(S). Discussiones Mathematicae. Graph Theory, Tome 35 (2015) no. 2, pp. 215-227. http://geodesic.mathdoc.fr/item/DMGT_2015_35_2_a1/