The Minimal Molecular Tree for the Exponential Randić Index

Topological indices are numerical parameters that provide a way to quantify the structural features of molecules using their graph representations. In chemical graph theory, these indices have been effectively employed to predict various physico-chemical properties of molecules. Among these, the Randić index stands out as a classical and widely used molecular descriptor in chemistry and pharmacology. The Randić index R ( G ) for a given graph G is defined as R ( G ) = ∑ v i v j ∈ E ( G ) 1 d ( v i ) d ( v j ) , where d ( v i ) represents the degree of vertex v i and E ( G ) is the set of edges in the graph G . Given the Randić index’s strong discrimination ability in describing molecular structures, a variant known as the exponential Randić index was recently introduced. The exponential Randić index E R ( G ) for a graph G is defined as E R ( G ) = ∑ v i v j ∈ E ( G ) e 1 d ( v i ) d ( v j ) . This paper further explores and fully characterizes the minimal molecular trees in relation to the exponential Randić index. Moreover, the chemical relevance of the exponential Randić index is also investigated.