Two problems on the Sombor indices of molecular graphs

Gutman proposed a new alternative interpretation of vertex-degree-based topological index, called Sombor index. It is defined via the term $\sqrt{d_G(v_i)^2+d_G(v_j)^2}$, where d G ( v i ) is the degree of the vertex v i in graph G . Deng et al. [Int. J. Quantum Chem. 121(11) (2021), e26622] proposed the following two problems: to determine the maximum value of the (reduced) Sombor index among all molecular graphs of order n and to determine the extremal (minimum and maximum) of the (reduced) Sombor index among all connected graphs with n vertices and m edges. In this paper, we solve these two problems. Keywords: Degree; Sombor index; Molecular tree AMS subject classification 2010: 05C05, 05C12, 05C35