The study of spectrum (i.e. multiset of eigenvalues) of graphs especially molecular graphs has found a considerable use in theoretical chemistry and computational nanoscience. There are various kinds of spectrum related topological indices such as, energy, Estrada index, Laplacian Estrada index, positive and negative inertia indices, nullity, signature etc., which play a key role in QSAR/QSPR studies. Energy and Estrada indices relate energies associated to the π -electron orbitals with the conjugated hydrocarbons and the degree of folding of proteins and some other long-chain biomolecules respectively. Whereas, inertia indices and nullity have found a significant use in the study of stability of chemical compounds. In this paper, inertia indices, nullity and signature of CNC3[n] and CNC5[n] nanocones are studied. Furthermore, we formulate a conjecture on these numerical parameters for general CNC2k+1[n] nanocones