Reciprocal complementary distance spectra and reciprocal complementary distance energy of line graphs of regular graphs

The reciprocal complementary distance (RCD) matrix of a graphG is defined asRCD(G) = [rcij] where rcij = 1 1+D−dij if i 6= j and rcij = 0, otherwise, where D is the diameter of G and dij is the distance between the vertices vi and vj 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 RCDequienergetic 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 RCDeigenvalues.