University of Kashan Iranian Journal of Mathematical Chemistry 2228-6489 17 3 2026 09 01 A Note on the General Gutman Index of Graphs 271 285 115735 10.22052/ijmc.2025.257853.2079 EN Zohreh Jamalkashani Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Kashan‎, ‎Kashan‎, ‎Iran ‎Gholam Hossein Fath-Tabar Department of Pure Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Kashan‎, ‎Kashan‎, ‎Iran 0000-0003-1105-3020 Mahdi Sabzevari Department of Applied Mathematics‎, ‎Faculty of Mathematical Sciences‎, ‎University of Kashan‎, ‎Kashan‎, ‎Iran Journal Article 2025 10 27 ‎The general Gutman index‎, ‎denoted by $G_{a,b}(H)$‎, ‎is a prominent distance–degree-based topological index in chemical graph theory‎, ‎defined using a combination of vertex degrees and intervertex distances‎. ‎In this paper‎, ‎we focus on bicyclic graphs as a special class of graphs and investigate the behavior of this index within that setting‎. ‎In particular‎, ‎we examine the case $G_{2,1}(H)$‎. ‎The primary objective of this study is to establish a sharp upper bound for the general Gutman index in bicyclic graphs‎. ‎By exploiting the structural properties of bicyclic graphs and employing combinatorial techniques‎, ‎we derive a sharp upper bound for this index‎. ‎The results obtained may provide a useful foundation for further research on distance-degree-based indices and their structural implications in mathematical chemistry‎. https://ijmc.kashanu.ac.ir/article_115735_a99546f93372387485650d3aff7fc09a.pdf