Novel/Old Generalized Multiplicative Zagreb Indices of Some Special Graphs

Topological descriptor is a fixed real number directly attached with the molecular graph to predict the physical and chemical properties of the chemical compound. Gutman and Trinajsti  elaborated the first Zagreb index (ZI) which was based on degree in 1972. Ali and Trinajsti defined a connection number (CN) based topological descriptor in 2018. They found that the CN-based Zagreb indices have a greater chemical capability for thirteen physicochemical properties of octane isomers.For  the generalized ZI and the generalized first Zagreb connection index (ZCI) of a graph  is    and , where  and  are the degree and CN of the vertex in . In this paper, the generalized first, second, third, and fourth multiplicative ZCIs are defined. Some exact solutions are also developed for the generalized multiplicative ZI and the above-mentioned generalized multiplicative versions of some special graphs, which are flower, sunflower, wheel, helm, and gear. The results  and  are the generalized forms of the results  and  where,  respectively. d(m)^{\beta}+d(m)^{\alpha}d(l)^{\beta}]$ and $C_{\alpha,\beta}(Q)=\sum_{lm\in E(Q)}[\tau(l)^{\alpha}\tau(m)^{\beta}+\tau(m)^{\alpha}\tau(l)^{\beta}]$, where $d_{Q}(m)$ and $\tau_{Q}(m)$ are the degree and CN of the vertex $m$ in $Q$. In this paper, we define generalized multiplicative ZCIs such as generalized first multiplicative ZCI, generalized second multiplicative ZCI, generalized third multiplicative ZCI and generalized fourth multiplicative ZCI. We also compute some exact formulae for the generalized multiplicative ZI and above-mentioned generalized multiplicative versions of some special graphs which are wheel, gear, helm, flower and sunflower. The obtained results $(MC_{\alpha,\beta}^{2},MC_{\alpha,\beta}^{3}$ and $MC_{\alpha,\beta}^{4})$ are the generalized extensions of the results $(MZC_{1}^{*}, MZC_{2}^{*}$ and $MZC_{3}^{*})$ of Javaid et al. [Novel connection based ZIs of several wheel-related graphs, $2(2020), 31-58]$ who worked only for $\alpha, \beta=1,$ respectively.