Computing the forgotten topological index of four operations on graphs

Abstract For a (molecular) graph, the first Zagreb index M 1 is equal to the sum of squares of the degrees of vertices, and the second Zagreb index M 2 is equal to the sum of the products of the degrees of pairs of adjacent vertices. The F -index of a graph G denoted by F ( G ) or M 3 ( G ) is defined as the sum of cubes of the degrees of vertices of the graph. The total π -electron energy depends on the degree based sum M 1 ( G ) = ∑ v ∈ V deg G ( v ) 2 and F ( G ) = ∑ v ∈ V deg G ( v ) 3 , it was shown in the study of structure-dependency of total π -electron energy in 1972. The first index was named first Zagreb index and the second sum ∑ v ∈ V deg G ( v ) 3 has been never further studied. Recently, this sum was named Forgotten index or the F-index by Furtula and Gutman and it was shown to have an exceptional applicative potential. The first and second Zagreb indices for the four operations on graphs were studied by Deng et al. (2016). In this paper, we extend this study to the F -index of graphs and determine the closed formulas for the F -index of four operations on graphs.