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Computing the forgotten topological index of four operations on graphs. (English) Zbl 1372.05038
Summary: 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 $$\pi$$-electron energy depends on the degree based sum $$M_1(G) = \sum_{v \in V} \deg_G(v)^2$$ and $$F(G) = \sum_{v \in V} \deg_G(v)^3$$, it was shown in the study of structure-dependency of total $$\pi$$-electron energy in [I. Gutman and N. Trinajstić, “Graph theory and molecular orbitals. Total $$\pi$$-electron energy of alternant hydrocarbons”, Chem. Phys. Lett. 17, No. 4, 535–538 (1972; doi:10.1016/0009-2614(72)85099-1)]. The first index was named first Zagreb index and the second sum $$\sum_{v \in V} \deg_G(v)^3$$ has been never further studied. Recently, this sum was named forgotten index or the F-index by B. Furtula and I. Gutman [J. Math. Chem. 53, No. 4, 1184–1190 (2015; Zbl 1317.05031)] 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 H. Deng et al. [“The Zagreb indices of four operations on graphs”, Appl. Math. Comput. 275, 422–431 (2016; doi:10.1016/j.amc.2015.11.058)]. 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.

##### MSC:
 05C07 Vertex degrees 05C76 Graph operations (line graphs, products, etc.)
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##### References:
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