Reciprocal complementary distance spectra and reciprocal complementary distance energy of line graphs of regular graphs - Harishchandra S. Ramane; Ashwini S. Yalnaik (GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB, 2015)
The reciprocal complementary distance (RCD) matrix of a graph G is defined as RCD(G) = [rc_ij] where rc_ij = 11+D-d_ij if i ≠ j and rc_ij = 0, otherwise, where D is the diameter of G and d_ij is the distance between the vertices v_i and v_j in G. The RCD-energy of G is defined as the sum of the absolute values of the eigenvalues of RCD(G). Two graphs are said to be RCD-equienergetic if they have same RCD-energy. In this paper we show that the line graph of certain regular graphs has exactly one positive RCD-eigenvalue. Further we show that RCD-energy of line graph of these regular graphs is solely depends on the order and regularity of G. This results enables to construct pairs of RCD-equienergetic graphs of same order and having different RCD-eigenvalues.