F-index of graphs based on new operations related to the join of graphs

Let G be a simple graph with the vertex set V(G) and edge set E(G) respectively. The degree of a vertex v in G is the number of vertices in G which are connected to v by an edge and denoted by dG(v). A topological index is a graph invarient which is numerical parameter obtained from a graph which characterize its topology.Thus for two isomorphic graphG and H the value of a particular topological index must be same for both of them. In graph theory, there are many topological indices which have very useful applications in Chemistry, Bio-Chemistry, Molecular Biology, Nanotechnology for QSAR/QSPR investigation, isomer discrimination, pharmaceutical drug design and much more. The most thoroughly studied and oldest topological indices are first and second Zagreb indices which were introduced by Gutman and Trinajestić [1] in a paper in 1972 to study structure dependency of the total π-electron energy(ǫ). They are respectively defined