The Harary Index is an important topological parameter for examining the structure of a graph. This work presents a quantitative analysis of the structural features of zero-divisor graph using the Harary Index. In this research article, we compute the Harary Index for the zero-divisor graph of the ring of upper triangular matrices over a finite commutative ring and non commutative ring. In this study, we have computed the Harary index for directed and undirected zero-divisor graph obtained from upper triangular matrix \ ( (M₃) \).
Alshanqiti et al. (Wed,) studied this question.