Zagreb Indices and Adjacency Matrices of Fuzzy Zero Divisor Graphs with MATLAB Based Algebraic Applications

Abstract This article aims to explore the theoretical properties of topological descriptors used in fuzzy graph theory, which combines elements of graph theory and fuzzy set theory, within algebraic structures. For this purpose, fuzzy Zagreb topological indices have been formulated theoretically and also adjacency matrices are constructed for the fuzzy zero-divisor graphs of the commutative ring Zn where n=pα , n=p1 p2 (p1 and p2 are primes). In particular, a SageMath-based drawing algorithm that embodies the fuzzy graph structures of the rings is presented for application-based convenience. Furthermore, a MATLAB-based code was created that directly calculates the Zagreb indices and adjacency matrices of the fuzzy zero-divisor graphs for every conceivable value of $n$. And thanks to this code, applications are included where we can calculate the characteristic polynomials, eigenvalues and graph energies of the fuzzy algebraic graphs we are working on. AMS Classification: 05C25, 05C72, 05C09