Abstract
Topological index (TI) is a function that assigns a numeric value to a (molecular) graph that predicts its various physical and structural properties. In this paper, we study the sum graphs (S-sum, R-sum, Q-sum and T-sum) using the subdivision related operations and strong product of graphs which create hexagonal chains isomorphic to many chemical compounds. Mainly, the exact values of first general Zagreb index (FGZI) for four sum graphs are obtained. At the end, FGZI of the two particular families of sum graphs are also computed as applications of the main results. Moreover, the dominating role of the FGZI among these sum graphs is also shown using the numerical values and their graphical presentations.
Highlights
Let G (V(G), E(G)) be a graph with V(G) and E(G) as sets of vertex and edge respectively. e degree of a vertex vεV(G) is the number of edges which are incident to v
topological index (TI) are used to predict the various chemical and structural properties such as octane isomers including entropy, acentric factor, density, total surface area, molar volume, boiling point, capacity of heat at temperature and pressure, enthalpy of formation, connectivity of compounds and octanol water partition, see [4]. ese are used to study the quantitative structure property and activity relationships which are very important in the subject of cheminformatics, see [5]
Gutman and Trinajstic see [7] investigated total π-electron energy of the molecule graphs using a degree-based TI called by first Zagreb index nowadays
Summary
Let G (V(G), E(G)) be a (molecular) graph with V(G) and E(G) as sets of vertex and edge respectively. e degree of a vertex vεV(G) is the number of edges which are incident to v. Eliasi and Taeri [12] constructed the Φ-sum graphs GΦ + H by the Cartesian product of the graphs Φ(G) and H, where Φ(G) is obtained after applying the Φ on G for Φ ∈ {S, R, Q, T} They computed the Wiener indices of the Φ-sums graphs GS + H, GR + H, GQ + H and GT + H. Akhter and Imran [14] computed the forgotten topological index of four operations on graphs under Cartesian product. We extend this study and compute FGZI of the Φ-sums graphs (GΦ ⊠ H) under the operation of strong product of Φ(G) and H in term of FGZI of its factor graphs G and H, where Φ ∈ {S, R, Q, T}, G and H are any two connected graphs. Journal of Mathematics are arranged as: Section 2 includes the basic formulae and results, Section 3 covers the main results and Section 4 includes the conclusion
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