The Minimum Reduced Sombor Index of Unicyclic Graphs in Terms of the Girth

Aims: The paper investigates the Reduced Sombor Index () for unicyclic graphs. Specifically, it aims to determine and characterize the unicyclic graphs that attain the minimum index among all unicyclic graphs of a given order. Study Design: This is a theoretical mathematical study based on graph theory and topological indices. The study involves defining and analyzing the Reduced Sombor Index by comparing values across different unicyclic graphs. Lemmas and theorems are proved to establish the minimum index graph. Methodology: Several graph transformation operations are analyzed. The study proves multiple lemmas that compare values before and after transformations by demonstrating whether a specific structural modification increases or decreases the value. Results: The minimum value in unicyclic graphs is achieved only by cycle graphs. Several lemmas prove that adding pendant vertices or modifying graph structure increases RSO. The final theorem establishes that for any unicyclic graph of order n, with equality if and only if G is a cycle . Conclusion: The study successfully characterizes unicyclic graphs with the minimum Reduced Sombor Index ). It establishes that cyclic graphs are the unique minimizers of the index among unicyclic graphs. Any structural modification leading to non-cycle unicyclic graphs increases . The findings contribute to chemical graph theory by refining how topological indices behave in molecular graph models.