# Adjacency matrix of total graph

Is there a nice way of relating the adjacency, incidence , Laplacian matrices and other matrices associated to a graph of a total graph with its original graph, or, say, at least relating that of the line graph with the graph?

I think there must be some relation. I also think the rank of the Laplacian of the graph and that of the graph also are closely related. Are there any references for this? Thanks beforehand.

If $$C$$ is the incidence matrix (rows indexed by vertices, columns by edges, two 1s in each column) then $$CC^T$$ is the Laplacian matrix and \$C^TC-2I is the adjacency matrix of the linegraph. From this it follows that the non-zero eigenvalues of the Laplacian are the same as the non-zero (eigenvalues + 2) of the linegraph.