The only finite connected graphs $G$ that are isomorphic to their [line graph](http://en.wikipedia.org/wiki/Line_graph) $L(G)$ are the [cycle graphs](http://en.wikipedia.org/wiki/Cycle_graph) $C\_n$ (see [this link](http://mathworld.wolfram.com/LineGraph.html) for example).


There are connected countable graphs that are isomorphic to their line graph:


* $G=(\omega,E)$ where $E=\{\{k,k+1\}: k\in\omega\}$;
* $G=(\mathbb{Z},E)$ where $E=\{\{k,k+1\}: k\in\mathbb{Z}\}$.


Note that in the second graph, all vertices have degree 2, which makes it into a kind of "infinite cycle". (An interesting side question would be whether these are the only connected countable graphs (up to isomorphism) that are isomorphic to their line graph.)


**Question.** Is there a connected graph $G=(V,E)$ with $V$ uncountable such that $G\cong L(G)$?



Infinite graphs isomorphic to their line graph
Note first that $L$ is naturally an endofunctor on the category of graphs and injective graph-homomorphisms that commutes with filtered colimits. Let $G$ be any graph such that there is an embedding $i:G\to L(G)$. This gives rise to an embedding $L(i):L(G)\to L(L(G))$, an embedding $L(L(i)):L(L(G))\to L(L(L(G)))$, and so on. Let $L^\omega(G)$ be the colimit of $G\to L(G)\to L(L(G))\to \dots$. Since $L$ commutes with filtered colimits, there are canonical isomorphisms $$L(L^\omega(G))\cong L(\varinjlim(G\to L(G)\to\dots))\cong \varinjlim( L(G)\to L(L(G))\to\dots)\cong L^\omega(G).$$


Furthermore, if $G$ is connected, so is $L^\omega(G)$, and if $G$ is infinite, $L^\omega(G)$ has the same cardinality as $G$. Thus to get a connected graph of a given infinite cardinality isomorphic to its line graph, it suffices to give a connected graph $G$ of that cardinality that embeds in $L(G)$. But this is easy; for instance, complete graphs work. In fact, any connected graph whose cardinality $\kappa$ has uncountable cofinality works, since such a graph must contain a vertex of degree $\kappa$, and thus $L(G)$ contains a clique of size $\kappa$.


