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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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