The line graph L of an undirected graph G is the graph whose vertex set is the edge set of the original graph G and in which two vertices are adjacent if their corresponding edges share a common endpoint in G.
i1 : G = graph({{1,2},{2,3},{3,4},{4,1},{1,3},{4,2}},EntryMode=>"edges")
o1 = Graph{1 => {2, 3, 4}}
2 => {1, 3, 4}
3 => {1, 2, 4}
4 => {1, 2, 3}
o1 : Graph
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i2 : lineGraph G
o2 = Graph{set {1, 2} => {set {1, 3}, set {1, 4}, set {2, 3}, set {2, 4}}}
set {1, 3} => {set {1, 2}, set {1, 4}, set {2, 3}, set {3, 4}}
set {1, 4} => {set {1, 2}, set {1, 3}, set {2, 4}, set {3, 4}}
set {2, 3} => {set {1, 2}, set {1, 3}, set {2, 4}, set {3, 4}}
set {2, 4} => {set {1, 2}, set {1, 4}, set {2, 3}, set {3, 4}}
set {3, 4} => {set {1, 3}, set {1, 4}, set {2, 3}, set {2, 4}}
o2 : Graph
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The object lineGraph is a method function.