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

Abstract

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 [Formula: see text] is weighted by a local bond factor [Formula: see text] 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 [Formula: see text]. This framework allows us to derive closed-form expressions for all sixteen modified bond-based indices on a broad collection of benchmark families: paths [Formula: see text], cycles [Formula: see text], complete graphs [Formula: see text], complete bipartite graphs [Formula: see text], stars [Formula: see text], friendship graphs [Formula: see text], wheels [Formula: see text], book graphs [Formula: see text], Dutch windmill graphs [Formula: see text], and hypercubes [Formula: see text]. 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 [Formula: see text], 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 [Formula: see text] 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 [Formula: see text]; Global zRMSE [Formula: see text]), improving upon the Physicochemical descriptors scenario (Macro Test [Formula: see text]; Global zRMSE [Formula: see text]).