On the Distance Spectrum and Distance-Based Topological Indices of Central Vertex-Edge Join of Three Graphs

In this paper, we introduce a new graph operation based on a central graph called central vertex-edge join (denoted by $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$) and then determine the distance spectrum of $G_{n_1}^C \triangleright (G_{n_2}^V\cup G_{n_3}^E)$ in terms of the adjacency spectra of regular graphs $G_1$, $G_2$ and $G_3$ when $G_1$ is triangle-free. As a consequence of this result, we construct new families of non-D-cospectral D-equienergetic graphs. Moreover, we determine bounds for the distance spectral radius and distance energy of the central vertex-edge join of three regular graphs. In addition, we provide 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, Wiener index. Using these results, we calculate the topological indices of the organic compounds Methylcyclobutane $(C_5H_{10})$ and Spirohexane $(C_6H_{10})$.