# Degree distribution of the line graph of an Erdös-Rényi random graph

An Erdös-Rényi random graph is a graph, which consists of N nodes and where each link between them is present with probability p. It comes natural then that the pdf giving the probability of a node in that gaph having k neighbours is a binomial.

The definition of a line graph is:

A line graph L(G) of a graph G is obtained by associating every link of G to a node of L(G) and connecting two nodes of L(G) iff the correspoding edges of G have a node.

Question:

If the degree distribution of an arbitrary E-R random graph G is binomial, as mentioned before, what would be the the degree distribution of its corresponding line graph L(G).

Fix an edge $e=\{u,w\}$ of $G$ and denote the vertex of $L(G)$ associated to it by $v_e$. By the definition of $L(G)$ should be clear that $$d_{L(G)}(v_e) = k \iff d_G(u)+d_G(w) = k+2$$ The reason for 2 at the right hand side of $d_G(u)+d_G(w) = k+2$ is that, if $e$ exists, then $u$ and $w$ already have degree at least one. But the connections which really contribute to the degree of $v_e$ are those made using other vertices but $u$ and $w$. Thus we have to subtract 1 from $d_G(u)$ and $d_G(w)$.
Consequently $d_{L(G)}(v_e) \sim \text{bin}(2n-4,p)$