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In the mathematical discipline of graph theory, the line graph of an undirected graph ''G'' is another graph ''L''(''G'') that represents the adjacencies between edges of ''G''. The name line graph comes from a paper by although both and used the construction before this. Other terms used for the line graph include the covering graph, the derivative, the edge-to-vertex dual, the conjugate, the representative graph, and the ϑ-obrazom, as well as the edge graph, the interchange graph, the adjoint graph, and the derived graph.〔, p. 71.〕 proved that with one exceptional case the structure of a connected graph ''G'' can be recovered completely from its line graph.〔 Many other properties of line graphs follow by translating the properties of the underlying graph from vertices into edges, and by Whitney's theorem the same translation can also be done in the other direction. Line graphs are claw-free, and the line graphs of bipartite graphs are perfect. Line graphs can be characterized by nine forbidden subgraphs, and can be recognized in linear time. Various generalizations of line graphs have also been studied, including the line graphs of line graphs, line graphs of multigraphs, line graphs of hypergraphs, and line graphs of weighted graphs. ==Formal definition== Given a graph ''G'', its line graph ''L''(''G'') is a graph such that * each vertex of ''L''(''G'') represents an edge of ''G''; and * two vertices of ''L''(''G'') are adjacent if and only if their corresponding edges share a common endpoint ("are incident") in ''G''. That is, it is the intersection graph of the edges of ''G'', representing each edge by the set of its two endpoints.〔 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Line graph」の詳細全文を読む スポンサード リンク
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