Complementary Distance Spectra and Energy of Certain Classes of Graphs

For a connected graph G, the CD-eigenvalues are the eigenvalues of its complementary distance matrix $$CD_{G}.$$ The sum of the absolute values of CD-eigenvalues gives the CD-energy $${\mathcal {E}}_{CD}(G)$$ of G. This study focuses on l-fold and strong l-fold graphs, where we investigate their CD-eigenvalues. Moreover, the relationships between the CD-energies of the l-fold graphs and strong l-fold graphs concerning the CD-energy of the original graph G are established. Notably, the CD-energy of these graph types exhibits dependence on the diameter of original graph involved. Also, explore the connection between the CD-energy of extended bipartite double cover graphs derived from certain regular graphs and their adjacency energy. Furthermore, the instances of graphs that demonstrate equienergetic properties concerning the adjacency and complementary distance matrices are presented.