Harary spectra and Harary energy of line graphs of regular graphs

Abstract

The Harary matrix of a graph G is defined as RD(G) = [r_ij] in which r_ij = 1 ∕ d_ij if i ≠ j and r_ij = 0 if i = j, where d_ij is the distance between the vertices v_i and v_j in G. The Harary energy of G is defined as the sum of the absolute values of the eigenvalues of Harary matrix. Two graphs are said to be Harary equienergetic if they have same Harary energy. In this paper we show that the Harary matrix of complement of the line graph of certain regular graphs has exactly one positive eigenvalue. Further we obtain the Harary energy of line graphs and of complement of line graphs of certain regular graphs and thus constructs pairs of Harary equienergetic graphs of same order and having different Harary eigenvalues.