Reciprocal Complementary Distance Energy of Complement of Line Graphs of Regular Graphs

Abstract

The reciprocal complementary distance (RCD) matrix of a graph G is defined as RCD(G) = [r ij ], where r ij = 1 1+D−dij if i = j and r 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-matrix. Two graphs are said to be RCD-equienergetic if they have same RCD-energy. In this paper, the RCD-energy of the complement of line graphs of certain regular graphs in terms of the order and degree is obtained and as a consequence, pairs of RCD-equienergetic graphs of same order and having different RCD-eigenvalues are constructed.