COMPLEMENTARY DISTANCE SPECTRA AND COMPLEMENTARY DISTANCE ENERGY OF LINE GRAPHS OF REGULAR GRAPHS

The complementary distance (CD) matrix of a graph G is defined as CD(G) = [c(ij)], where c(ij) = 1 + D - d(ij) if i not equal j and c(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 CD-energy of G is defined as the sum of the absolute values of the eigenvalues of CD-matrix. Two graphs are said to be CD-equienergetic if they have same CD-energy. In this paper we show that the complement of the line graph of certain regular graphs has exactly one positive CD-eigenvalue. Further we obtain the CD-energy of line graphs of certain regular graphs and thus constructs pairs of CD-equienergetic graphs of same order and having different CD-eigenvalues.