Computing Zagreb Indices of the Subdivision-Related Generalized Operations of Graphs

Mathematical modeling or numerical coding of the molecular structures play a significant role in the studies of the quantitative structure-activity relationships (QSAR) and quantitative structures property relationships (QSPR). In 1972, the entire energy of
$\pi $
-electron of a molecular graph is computed by the addition of square of degrees (valencies) of its vertices (nodes). Later on, this computational result was called by the first Zagreb index and became well studied topological index in the field of molecular graph theory. In this paper, for
$k\in N$
(set of counting numbers), we define four subdivision-related operations of graphs in their generalized form named by
$S_{k}$
,
$R_{k}$
,
$Q_{k}$
and
$T_{k}$
. Moreover, using these operations and the concept of the cartesian product of graphs, we construct the generalized
$F_{k}$
-sum graphs
$\Gamma _{1 +F_{k}} \Gamma _{2}$
, where
$F_{k}\in \{S_{k}, R_{k}, Q_{k}, T_{k}\}$
and
$\Gamma _{i}$
are any connected graphs for
$i\in \{1,2\}$
. Finally, the first and second Zagreb indices are computed for the generalized
$F_{k}$
-sum graphs in terms of their factor graphs. In fact, the obtained results are a general extension of the results Eliasi
et al.
and Deng
et al.
who studied these operations for exactly
$k=1$
and computed the Zagreb indices for only
$F_{1}$
-sum graphs respectively.