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

A graph with q ( a + t ) vertices is known as a q -broom-like graph K q ⊓ B ( a ; t ) , which is produced by the hierarchical product of the complete graph K q by the rooted broom B ( a ; t ) , where q ≥ 3 , a ≥ 1 and t ≥ 1 . A numerical quantity associated with graph structure is called a topological index. The inverse sum indeg index (shortened to I S I index) is a topological index defined as I S I ( G ) = ∑ v i v j ∈ E ( G ) d v i d v j d v i + d v j , where d v i is the degree of the vertex v i . In this paper, we take into consideration the I S I index for q -broom-like graphs and perform a thorough analysis of it. We find the I S I spectrum of q -broom-like graphs and derive the closed formulas for their I S I index and I S I energy. We also characterize extremal graphs and arrange them according to their I S I index and I S I energy, respectively. Further, a quantitative structure–property relationship is used to predict six physicochemical properties of sixteen alkaloid structures using I S I index and I S I 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.