Abstract
The Sombor index, introduced by Ivan Gutman in 2020, has received intensive attention. The Sombor index of a graph \(G\) is defined as \(SO(G)=\sum\limits_{uv\in E(G)}\sqrt{d_{u}^{2}+d_{v}^{2}}\), where \(E(G)\) denotes the edge set in \(G\) and \(d_{u}\) denotes the degree of vertex \(u\) in \(G\). A graph with maximum degree at most 4 is called as a chemical graph.
Réti et al. [T. Réti, T. Došlić, A. Ali, On the Sombor index of graphs, Contrib. Math. 3 (2021) 11-18] proposed an open problem about determining the maximum Sombor index among all connected \(c\)-cyclic graph for \(6\leq c\leq n-2\). For \(c=1,2,3,4\), the problem about finding the minimum (resp. maximum) Sombor index among all connected \(c\)-cyclic graph has already been solved. In this paper, we determine the minimum Sombor index among connected \(c\)-cyclic chemical graph for \(c\geq 3, n\geq 5(c-1)\), which partially extends the results of Liu et al. [H. Liu, L. You, Y. Huang, Ordering chemical graphs by Sombor indices and its applications, MATCH Commun. Math. Comput. Chem. 87 (2022) 5-22] and Liu et al. [H. Liu, L. You, Y. Huang, Extremal Sombor indices of tetracyclic (chemical) graphs, MATCH Commun. Math. Comput. Chem. 88 (2022) 573-581].