A study on $k$-coalescence of two graphs

The $k$-coalescence of two graphs is obtained by merging a $k$-clique of each graph. The $A_\alpha$-matrix of a graph is the convex combination of its degree matrix and adjacency matrix. In this paper, we present some structural properties of a non-regular graph which is obtained from the $k$-coalescence of two graphs. Also, we derive the $A_\alpha$-characteristic polynomial of $k$-coalescence of two graphs and then compute the $A_\alpha$-spectra of $k$-coalescence of two complete graphs. In addition, we estimate the $A_\alpha$-energy of $k$-coalescence of two complete graphs. Furthermore, we obtain some topological indices of vertex coalescence of two graphs, and as an application, we determine some indices of some family of graphs. From these results, we calculate the Wiener index, hyper-Wiener index etc. of the organic compound 1,2-dicyclohexylethane(\ce{C_{14}H_{26}}).