Combinatorial approach to structural indices of zero-divisor graphs

In this paper, we offer a simple combinatorial approach for calculating the distance matrix, distance-based structural polynomials and many distance-based indices of the zero-divisor graph [Formula: see text] associated to a finite reduced ring [Formula: see text] where [Formula: see text] is a finite field. A key contribution is the development of an efficient algorithm that computes the exponentially large-sized distance matrix of [Formula: see text] in terms of smaller-sized variants of the distance matrix. By utilizing the distance matrix, we derive explicit formulas for important distance-based graph invariants of [Formula: see text] such as the Hosoya polynomial, Harary polynomial, and Harary index, Wiener index and its variants. We achieve this by formulating recurrence relations using the distance matrix and solving them combinatorially. Consequently, for the Boolean graph and the zero-divisor graph [Formula: see text] ([Formula: see text] a prime), the distance matrix and the distance-based invariants above can be explicitly obtained. Moreover, we establish an upper bound for the Wiener index of [Formula: see text] and also show that this bound is attained if and only if [Formula: see text] is prime.