The LineGraph command creates the line graph $L\left(G\right)$ of a given graph G. The vertices of $L\left(G\right)$ are the edges of G and two vertices are connected by an edge if the corresponding edges in G are adjacent.

$\mathrm{with}\left(\mathrm{GraphTheory}\right)\:$

$\mathrm{K4}≔\mathrm{CompleteGraph}\left(4\right)$

$\mathrm{K4}≔\mathrm{Graph\; 1:\; an\; undirected\; unweighted\; graph\; with\; 4\; vertices\; and\; 6\; edge(s)}$

$\mathrm{DrawGraph}\left(\mathrm{K4}\right)$

$L≔\mathrm{LineGraph}\left(\mathrm{K4}\right)$

$L≔\mathrm{Graph\; 2:\; an\; undirected\; unweighted\; graph\; with\; 6\; vertices\; and\; 12\; edge(s)}$

$\mathrm{DrawGraph}\left(L\right)$

$\mathrm{Vertices}\left(L\right)$

$\left[''1-2''\,''1-3''\,''1-4''\,''2-3''\,''2-4''\,''3-4''\right]$

$\mathrm{Edges}\left(\mathrm{GraphComplement}\left(L\right)\right)$

$\left\{\left\{''1-2''\,''3-4''\right\}\,\left\{''1-3''\,''2-4''\right\}\,\left\{''1-4''\,''2-3''\right\}\right\}$