Spectra and energies of iterated line graphs of regular graphs

If G is a graph and L(G)=L1(G) its line graph, then Lk(G), k=2,3,…, defined recursively via Lk(G)=L(Lk−1(G)), are the iterated line graphs of G. If G is a regular graph of degree r, r≥3, then all negative eigenvalues of its iterated line graphs are equal to minus 2. The energy E(G) of a graph G is the sum of absolute values of the eigenvalues of G. If G is a regular graph of order n and of degree r≥3, then for each k≥2, E(Lk(G)) depends solely on n and r. In particular, E(L2(G))=2nr(r−2). This result enables a systematic construction of pairs of non-cospectral connected graphs of the same order, having equal energies.