  Topological indices are molecular descriptors that describe the properties of
chemical compounds. These topological indices correlate specific
physico-chemical properties like boiling point, enthalpy of vaporization,
strain energy, and stability of chemical compounds. This article introduces a
new graph operation based on central graph called central vertex-edge join and
provides its results related to graph invariants like eccentric-connectivity
index, connective eccentricity index, total-eccentricity index, average
eccentricity index, Zagreb eccentricity indices, eccentric geometric-arithmetic
index, eccentric atom-bond connectivity index, and Wiener index. Also, we
discuss the distance spectrum of the central vertex-edge join of three regular
graphs. Furthermore, we obtain new families of $D$-equienergetic graphs, which
are non $D$-cospectral.
