From graph theory to chemoinformatics: modified bond-based indices and a hypothesis-driven multi-task QSAR/QSPR benchmark

Graph–theoretic degree–based descriptors play a central role in chemoinformatics and QSPR/QSAR modelling, yet most classical indices either focus purely on vertex degrees or treat bond contributions in a purely multiplicative way. In this work we introduce and systematically study a new family of modified bond-based indices in which each edge \(uv∈ E(G)\) is weighted by a local bond factor \(T(uv)={φ _G(u)}+{φ _G(v)}-2\) in the denominator , coupled with a vertex kernel in the numerator. This construction yields modified versions of the first and second Zagreb indices, the Forgotten and Yemen indices, several connectivity-type descriptors (product, sum, Nirmala, ABC, CAB, GA, harmonic, and misbalance prodeg), as well as Sombor- and Dharwad-type bond indices. We first present a unified edge–partition representation for any symmetric kernel, expressing each modified index as a finite sum over degree classes \(E(a,b)(G)\) . This framework allows us to derive closed-form expressions for all sixteen modified bond-based indices on a broad collection of benchmark families: paths \(Pₙ\) , cycles \(Cₙ\) , complete graphs \(Kₙ\) , complete bipartite graphs \(Km,n\) , stars \(Sₙ\) , friendship graphs \(Fₙ\) , wheels \(Wₙ\) , book graphs \(Bₙ\) , Dutch windmill graphs \(Dₙ⁽ᵐ⁾\) , and hypercubes \(Q_d\) . The resulting tables reveal clear asymptotic growth patterns and highlight which structures are extremal for the modified descriptors. Moreover, we obtain sharp degree–extreme bounds for a representative subset of the indices in terms of the order \(n\) , size m , and the minimum and maximum degrees \(δ\) and \(Δ\) , with equality characterizing regular graphs. The proposed modified bond-based indices thus provide a flexible and analytically tractable family of descriptors that couple vertex and bond information in a novel way, and are well suited as structured features for modern chemoinformatics and graph-based machine-learning models on molecular graphs. Finally, to demonstrate predictive utility in a hypothesis-driven setting, we further benchmark these \({}ᵐBI\) descriptors within a large multi-task QSAR/QSPR pipeline on 3,219 ChEMBL antibacterial molecules across ten continuous properties using a heterogeneous model zoo under three descriptor scenarios, where the combined descriptors scenario achieves the best overall generalisation (Macro Test \(R^2 = 0.861\) ; Global zRMSE \(= 0.373\) ), improving upon the Physicochemical descriptors scenario (Macro Test \(R^2 = 0.852\) ; Global zRMSE \(= 0.385\) ).