On the Second Zagreb Index of Triangular Cycle Graphs

Abstract

This study investigates the second Zagreb index 2 M G( ) of triangular cycle graphs, derived from the geometric arrangement of triangular numbers. The second Zagreb index, defined as the sum of the products of vertex degrees for adjacent vertices, provides insights into edge-based connectivity and branching patterns. By analyzing the structure of triangular cycle graphs, we classify edges based on vertex degrees ( 2 2  , 2 3  , or 3 3  ) and derive explicit formulas for the second Zagreb index. Our results, including the formula 2 2 M T n n n ( ( )) 2 20 37    for n  3 , enhance the understanding of these graphs' topological properties and their applications in network analysis and chemical graph theory.