Abstract
Topological index (TI) is a function from the set of graphs to the set of real numbers that associates a unique real number to each graph, and two graphs necessarily have the same value of the TI if these are structurally isomorphic. In this note, we compute theHZ−indexof the four generalized sum graphs in the form of the various Zagreb indices of their factor graphs. These graphs are obtained by the strong product of the graphsGandDkG, whereDk∈Sk,Rk,Qk,Tkrepresents the four generalized subdivision-related operations for the integral value ofk≥1andDkGis a graph that is obtained by applyingDkonG. At the end, as an illustration, we compute theHZ−indexof the generalized sum graphs for exactlyk=1and compare the obtained results.
Highlights
A structural formula of a chemical compound is represented by a molecular graph, where atoms and bonds between atoms are represented by the vertices and edges of the molecular graphs, respectively
A topological index (TI) is a mathematical tool which associates a real number to a graph under certain conditions
Shirdel et al [22] studied the concept of hyper-Zagreb index (HZ − index) as a particular case of the general sum connectivity (GSC) index
Summary
A structural formula of a chemical compound is represented by a molecular graph, where atoms and bonds between atoms are represented by the vertices and edges of the molecular graphs, respectively. Trinajstic and Gutman (1972) [13] defined the first and second Zagreb indices that are used to compute the different structure base characteristics of the molecular graphs. Using the concept of Cartesian product of graphs, they constructed the generalized sum graphs or Dk-sum graphs (denoted by GDk + H) and computed their first and second Zagreb indices. Javaid et al (2021) [33] redefined these graphs using strong product and computed their Zagreb indices (first and second) In this development, we compute hyper-Zagreb indices (HZ − index) for these graphs in terms of various degree-based TIs of their factor graphs, where these generalized sum graphs are obtained with the help of strong product.
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