Reciprocal Complement Distance Eigenvalues and Energy of Some Classes of Graphs

For a graph Γ that is connected, the RCD-eigenvalues are defined as the eigenvalues of its reciprocal complementary distance matrix RCD(Γ). The sum of the absolute values of these eigenvalues is called the RCD-energy, denoted by ℰRCD(Γ). Two connected graphs of the same order with equal RCD-energy are said to be RCD-equienergetic. In this work, we study the RCD-spectra of l-fold and strong l-fold graphs. We establish relationships between the RCD-energies of the l-fold graphs and strong l-fold graphs concerning the RCD-energy of the original graph Γ. Notably, the RCD-energy of these graph types exhibits dependence on the diameter of the original graph involved. Furthermore, we present instances of graphs that demonstrate equienergetic properties concerning the adjacency and reciprocal complementary distance matrices.