Abstract:
One fascinating area of graph theory that has numerous applications in chemistry is
called chemical graph theory. A molecular graph is a simple graph in which vertices and
edges, respectively, represent atoms and the chemical bonds between them. Computing
topological indices is now one of the most active areas of chemical graph theory. The
main role of a topological index is to predict various physicochemical properties of a
molecule graph. Decks of cards, computer graphics, and other real-world objects can
all be represented using Cartesian products. This study focuses on finding topological
indices of the Cartesian product of Firecracker with P2 Path graph. A firecracker graph
(Fn,k) is created by concatenating n copies of k stars from the root of exactly one star
where n ≥ 2 and k ≥ 4. The Cartesian product of Firecracker with P2 Path graph
(Fp) is constructed by connecting two firecracker graphs Fn,k(n ≥ 2, k ≥ 4) using
the P2 graph. The Fp has order p and size p + nk − 2 where p = 2nk. E. Deutsch
and S. Klavˇzar introduced M-polynomials to determine the most general polynomials
to produce degree-based topological indices. The most important benefit of the Mpolynomial
is its wealth of information about degree-based graph transformations. The
closed form of many degree-based topological indices has been derived for the line graph
of the firecracker graph using an M-polynomial. In this study, we derived formulas for
the topological indices such as the First Zagreb Index, Second Zagreb Index, Third
Zagreb Index, Hyper - Zagreb Index, Randi´c Index, Atom-Bond Connectivity Index,
Geometric-Arithmetic Index, Sum-Connectivity Index, Zagreb Index, and Inverse Sum
Index of the Cartesian product of Firecracker with P2 Path graph using M-Polynomial.
