First general Zagreb index of generalized \(F\)-sum graphs (Q2657407)

Summary: The first general Zagreb (FGZ) index (also known as the general zeroth-order Randić index) of a graph \(G\) can be defined as \(M^\gamma\left( G\right)=\sum_{u v \in E \left( G\right)} \left[ d_G^{\gamma - 1} \left( u\right) + d_G^{\gamma - 1} \left( v\right)\right]\), where \(\gamma\) is a real number. As \(M^\gamma\left( G\right)\) is equal to the order and size of \(G\) when \(\gamma=0\) and \(\gamma=1\), respectively, \( \gamma\) is usually assumed to be different from 0 to 1. In this paper, for every integer \(\gamma\geq 2\), the FGZ index \(M^\gamma\) is computed for the generalized F-sums graphs which are obtained by applying the different operations of subdivision and Cartesian product. The obtained results can be considered as the generalizations of the results appeared in [\textit{J.-B. Liu} et al., ``Computing first general Zagreb index of operations on graphs'', IEEE Access 7, 47494--47502 (2019; \url{doi:10.1109/ACCESS.2019.2909822}); ``Computing Zagreb indices of the subdivision-related generalized opeations of graphs'', IEEE Access 7, 105479--105488 (2019; \url{doi:10.1109/ACCESS.2019.2932002})].