Abstract
The Sombor index indicated by the symbol \( SO(G) \) is calculated by adding the contributions of each vertex to the total number of edges in \( G \), while the reduced Sombor index \( SO_{red}(G) \) refines this measure by discounting the contributions of pendant vertices, which have a degree of 1.
\begin{align*}
SO(G) &= {\sum\limits_{xy\in E(G)}} \sqrt{{d_x}^2+{d_y}^2} \\
SO_{red}(G) &= {\sum\limits_{xy\in E(G)}} \sqrt{({d_x}-1)^2+({d_y}-1)^2}
\end{align*}
for a given vertex \( x \) in graph \( G \), \( {d_x} \) corresponds to the degree of that vertex. Our focus centers on exploring the Sombor index and reduced Sombor index of unicyclic graphs, specifically addressing graphs with a predetermined girth. We determine the first four smallest Sombor index and reduced Sombor index values and identifying the corresponding graphs that achieve these extremes.
