Sombor Index of c-Cyclic Chemical Graphs

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) = Sigma(uv is an element of E(G)) root 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. Reti et al. [T. Reti, T. Do.slic, 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 <= c <= 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 >= 3, n >= 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].