Generalized Reduced Gröbner Basis and Initial Ideal of Binomial Edge Ideal of Different Classes of Graphs

In this paper, we study the binomial edge ideals associated with three specific classes of graphs: the comb graph, the cross-ladder graph, and the -sunlet graph. These graph structures offer a rich interplay between combinatorics and algebra, particularly in the context of Gröbner basis theory. For each graph, we explicitly compute the reduced Gröbner basis of the corresponding binomial edge ideal with respect to a lexicographic monomial order. Our computations involve a detailed analysis of admissible paths in the graphs, which play a central role in characterizing the generators of the Gröbner basis. Furthermore, we determine the initial ideals associated with each class and describe the families of monomials that arise in their minimal generating sets. The construction of these Gröbner bases not only offers insight into the structural properties of the respective graphs but also enables potential applications in algebraic statistics, computational algebra, and ideal theory. By classifying the admissible paths and systematically generating the Gröbner basis elements, our work provides a constructive and combinatorially motivated framework for understanding binomial edge ideals. These results contribute to the growing body of literature exploring the connections between graph-theoretic structures and algebraic invariants, and they open avenues for further investigations in more generalized or complex graph families.