In this paper we find asymptotic enumerations for the number of line graphs
on n-labelled vertices and for different types of related combinatorial
objects called 2-covers.
We find that the number of 2-covers, snβ, and proper 2-covers, tnβ, on
[n] both have asymptotic growth snββΌtnββΌB2nβ2βnexp(β21βlog(2n/logn))=B2nβ2βn2nlognββ, where B2nβ is the 2nth Bell number, while the number of
restricted 2-covers, unβ, restricted, proper 2-covers on [n], vnβ, and
line graphs lnβ, all have growth unββΌvnββΌlnββΌB2nβ2βnnβ1/2exp(β[21βlog(2n/logn)]2).
In our proofs we use probabilistic arguments for the unrestricted types of
2-covers and and generating function methods for the restricted types of
2-covers and line graphs