Prove that $L=C^T C-2I$ 
The incidence matrix $C$ of a graph and adjacency matrix $L$ of its line graph are related by  $L=C^T C-2I$ 

I know this property  from this site. 
I understand how to do calculation by $L=C^T C-2I$ , but how to prove this equality. I hope someone can help me to understand clearly about that.
 A: The question you should ask (yourself) is:

What is the graph-theoretic interpretation of the matrix product $C^T C$?

Try to answer this, and you will get the answer. If you fail to find the answer, read further…
Notice that the $(i,j)$-entry of this product will be the "dot product" of the $i$-th and $j$-th columns of the incidence matrix $C$, which is, in some sense, a dot product of the $i$-th and $j$-th edges! (That is not a standard term at all, so you should forget that as soon as you finish reading this answer). What are the possible values of this dot product? Since the entries of each column are  $0$s and $1$s, with exactly two $1$s, the only possible values for the dot product are $0$, $1$, and $2$ (make sure you can see why this is so). To make it easier to see this, recall that the product is a sum of the form
\begin{equation*}
\large\sum_kc_{ki}c_{kj}.
\end{equation*}
Each term in the sum is either $0$ or $1$. It is $1$ if and only if both $c_{ki}$ and $c_{kj}$ are $1$, which happens exactly when both the $i$-th and $j$-th edges are incident with the $k$-th vertex (the rows of $C$ are indexed by vertices, and the columns by edges). In other words, each term in the sum representing the dot product of the $i$-th and $j$-th edges is $1$ exactly when they are both incident on a common vertices — and therefore, the dot product is exactly the number of vertices "shared" by the edges. The only possibilities for this are $0$, $1$, and $2$ (the last case being when $i = j$, in a simple graph).
Thus we see that $C^T C$ is a matrix with rows and columns indexed by the edges of the original graph $G$ (which become the vertices of the line graph, $\Lambda$), in which the $(i,j)$-entry is


*

*$0$ iff $i \ne j$ and the $i$-th and $j$-th edges of $G$ (vertices of $\Lambda$) are not adjacent

*$1$ iff the $i \ne j$ and the $i$-th and $j$-th edges of $G$ are adjacent

*$2$ iff $i = j$ (diagonal entry).


It follows immediately that $L = C^T C - 2I$ is the adjacency matrix of the line graph $\Lambda$.
