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 GCn1▹(GVn2∪GEn3)Gn1C▹(Gn2V∪Gn3E)) and then determine the distance spectrum of GCn1▹(GVn2∪GEn3)Gn1C▹(Gn2V∪Gn3E) in terms of the adjacency spectra of regular graphs G1G1, G2G2 and G3G3 when G1G1 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 (C5H10)(C5H10) and Spirohexane (C6H10)(C6H10).