D-Spectrum and D-Energy of Complements of Iterated Line Graphs of Regular Graphs

Abstract

The D-eigenvalues {µ₁,…,µ_p} of a graph G are the eigenvalues of its distance matrix D and form its D-spectrum. The D-energy, E_D(G) of G is given by E_D (G) =∑_i=1^p |µ_i|. Two non cospectral graphs with respect to D are said to be D-equi energetic if they have the same D-energy. In this paper we show that if G is an r-regular graph on p vertices with 2r ≤ p - 1, then the complements of iterated line graphs of G are of diameter 2 and that E_D(overline{L^k(G)}), k≥2 depends only on p and r. This result leads to the construction of regular D-equi energetic pair of graphs.