Spectra and energies of iterated line graphs of regular graphs

Abstract

If G is a graph and L ( G ) = L 1 ( G ) its line graph, then L k ( G ) , k = 2 , 3 , … , defined recursively via L k ( G ) = L ( L k − 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 ( L k ( G ) ) depends solely on n and r . In particular, E ( L 2 ( G ) ) = 2 n r ( r − 2 ) . This result enables a systematic construction of pairs of non-cospectral connected graphs of the same order, having equal energies.