Forgotten Topological Index of The Zero Divisor Graph for Some Rings of Integers

A topological index is a numerical value that provides information about the structure of a graph. Among various degree-based topological indices, the forgotten topological index (F-index) is of particular interest in this study. The F-index is calculated for the zero divisor graph of a ring R. In graph theory, the zero divisor graph of R is defined as a graph with vertex set the zero-divisors of R, and for distinct vertices a and b are adjacent if a · b = 0. This research focuses on the zero divisor graph of the commutative ring of integers modulo 2ρn where ρ is an odd prime and n is a positive integer. The objectives are to determine the set of all zero divisors, analyze the vertex degrees of the graph, and then compute the F-index of the zero divisor graph. Using algebraic techniques, we derive the degree of each vertex, the distribution of vertex degrees, and the number of edges in the graph. The general expression for the F-index of the zero divisor graph for the ring is established. The results contribute to understanding topological indices for algebraic structures, with potential applications in chemical graph theory and related disciplines.