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  "body": "\n## Definition\n\nLet $G = (V, E)$ be a [[graph|0JD5C82QMB]].\nThe line graph of $G$ is the graph $L(G)$ whose vertex set is the [[edge set|0WNGQBPPQX]] $E$ of $G$.\nTwo distinct edges $e, f \\in E$ are adjacent in $L(G)$ if and only if they have a common endpoint.\n\n## Remarks\n\nIf $e = uv$ is an edge of $G$, its degree in $L(G)$ is $\\deg u + \\deg v - 2$.\nExample: the [[cycle graph|0QQ0F12962]] $C_n$ with $n \\ge 3$ has $L(C_n) \\cong C_n$, since each edge meets exactly the two edges next to it around the cycle.\nAlso $L(K_{1,3}) \\cong K_3$: the three edges of the star share the centre, so they are pairwise adjacent.\nNon-example: the star $K_{1,3}$ is itself not the line graph of any graph.\nIndeed, suppose a vertex $e$ of $L(H)$ had three pairwise non-adjacent neighbours.\nThese would be three edges meeting $e$, so two of them would meet $e$ at the same endpoint, and these two would be adjacent.\nWhitney showed that two connected graphs with isomorphic line graphs are isomorphic, except for the pair $K_3$ and $K_{1,3}$.\n",
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