  Let $G$ be a graph with vertex set $V(G)$ and edge set $E(G)$. The Sombor and
reduced Sombor indices of $G$ are defined as $SO(G)=\sum_{uv\in
E(G)}\sqrt{deg_G(u)^2+deg_G(v)^2}$ and $SO_{red}(G)=\sum_{uv\in
E(G)}\sqrt{(deg_G(u)-1)^2+(deg_G(v)-1)^2}$, respectively. We denote by
$H_{n,\nu}$ the graph constructed from the star $S_n$ by adding $\nu$ edge(s)
$(0\leq \nu\leq n-2)$, between a fixed pendent vertex and $\nu$ other pendent
vertices. R\'eti et al. [T. R\'eti, T. Do\v{s}li\'c and A. Ali, On the Sombor
index of graphs, $\textit{Contrib. Math. }$ $\textbf{3}$ (2021) 11-18] proposed
a conjecture that the graph $H_{n,\nu}$ has the maximum Sombor index among all
connected $\nu$-cyclic graphs of order $n$, where $5\leq \nu \leq n-2$. In this
paper we confirm that the former conjecture is true. It is also shown that this
conjecture is valid for the reduced Sombor index. The relationship between
Sombor, reduced Sombor and first Zagreb indices of graph is also investigated.
