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In mathematical chemistry, molecular descriptors serve an important role, primarily in quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) studies. A topological index of a molecular graph is a real number that is invariant under graph isomorphism conditions and provides information about its size, symmetry, degree of branching, and cyclicity. For any graph N, the first and second Gourava indices are defined as G O 1 (N) = ∑ u ′ v ′ ∈ E (N) (d (u ′) + d (v ′) + d (u ′) d (v ′) ) and G O 2 (N) = ∑ u ′ v ′ ∈ E (N) (d (u ′) + d (v ′)) d (u ′) d (v ′) , respectively.The independence number of a graph N, being the cardinality of its maximal independent set, plays a vital role in reading the energies of chemical trees. In this research paper, it is shown that among the family of trees of order δ and independence number ξ , the spur tree denoted as Υ δ , ξ maximizes both 1st and 2nd Gourava indices, and with these characterizations this graph is unique. [ABSTRACT FROM AUTHOR]

In mathematical chemistry, molecular descriptors serve an important role, primarily in quantitative structure–property relationship (QSPR) and quantitative structure–activity relationship (QSAR) studies. A topological index of a molecular graph is a real number that is invariant under graph isomorphism conditions and provides information about its size, symmetry, degree of branching, and cyclicity. For any graph N, the first and second Gourava indices are defined as GO1(N)=∑u′v′∈E(N)(d(u′)+d(v′)+d(u′)d(v′)) and GO2(N)=∑u′v′∈E(N)(d(u′)+d(v′))d(u′)d(v′), respectively.The independence number of a graph N, being the cardinality of its maximal independent set, plays a vital role in reading the energies of chemical trees. In this research paper, it is shown that among the family of trees of order δ and independence number ξ, the spur tree denoted as Υδ,ξ maximizes both 1st and 2nd Gourava indices, and with these characterizations this graph is unique.