If $$C$$ is the incidence matrix (rows indexed by vertices, columns by edges, two 1s in each column) than $$CC^T$$ 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 name as the non-zero (eigenvalues + 2) of the linegraph.