Inverse Sum Indeg Spectrum of q-Broom-like Graphs and Applications

A graph with (Formula presented.) vertices is known as a q-broom-like graph (Formula presented.), which is produced by the hierarchical product of the complete graph (Formula presented.) by the rooted broom (Formula presented.), where (Formula presented.) and (Formula presented.). A numerical quantity associated with graph structure is called a topological index. The inverse sum indeg index (shortened to (Formula presented.) index) is a topological index defined as (Formula presented.), where (Formula presented.) is the degree of the vertex (Formula presented.). In this paper, we take into consideration the (Formula presented.) index for q-broom-like graphs and perform a thorough analysis of it. We find the (Formula presented.) spectrum of q-broom-like graphs and derive the closed formulas for their (Formula presented.) index and (Formula presented.) energy. We also characterize extremal graphs and arrange them according to their (Formula presented.) index and (Formula presented.) energy, respectively. Further, a quantitative structure–property relationship is used to predict six physicochemical properties of sixteen alkaloid structures using (Formula presented.) index and (Formula presented.) energy. Both graph invariants have significant correlation values, indicating the accuracy and utility of the findings. The conclusions made in this article can help chemists and pharmacists research alkaloids’ structures for applications in industry, pharmacy, agriculture, and daily life.