# Spectrum of line graph of regular graph

Definition:
Let $$G$$ be a graph, the line graph of $$G$$ denoted of $$L(G)$$ is defined as follows:
-The vertices of $$L(G)$$ are the edges of $$G$$
-Two vertices of $$L(G)$$ are adjacent iff their corresponding edges in $$G$$ are incident G.

Question:
It is easy to see that if $$G$$ is $$d$$-regular graph then $$L(G)$$ is $$(2d-2)$$-regular graph. Thus $$2d-2$$ is eigenvalue of $$L(G)$$.

Let $$G$$ be $$d$$-regular graph. Is there any relation between the eigenvalues of $$L(G)$$ and those of $$G$$?

If $G$ is $k$-regular with $n$ vertices and $m=nk/2$ edges, and if $\phi(M,t)$ is the characteristic polynomial of $M$, then by a result due to Sachs, $\phi(L(G),t) = (t+2)^{m-n}\phi(G,t+2-k).$ I am lifting this from page 19 of the 2nd edition of Biggs "Algebraic Graph Theory" (where you will find a proof).