Resolving the Open Problem by Proving a Conjecture on the Inverse Mostar Index for c-Cyclic Graphs

Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index,  (Formula presented.), is defined as (Formula presented.) where (Formula presented.) and (Formula presented.) represent the number of vertices closer to vertex u than v and closer to v than u, respectively, for an edge (Formula presented.). The inverse Mostar index problem has gained significant attention recently. In their work, Alizadeh et al. [Solving the Mostar index inverse problem, J. Math. Chem. 62 (5) (2024) 1079–1093] proposed the following open problem: “Which nonnegative integers can be realized as Mostar indices of c-cyclic graphs, for a given positive integer c?”. Subsequently, one of the present authors [On the inverse Mostar index problem for molecular graphs, Trans. Comb. 14 (1) (2024) 65–77] conjectured that, except for finitely many positive integers, all other positive integers can be realized as the Mostar index of a c-cyclic graph, where (Formula presented.). In this paper, we address the inverse Mostar index problem for c-cyclic graphs. Specifically, we construct infinitely many families of symmetric c-cyclic structures, thereby demonstrating a solution to the inverse Mostar index problem using an infinite family of such symmetric structures. By providing a comprehensive proof of the conjecture, we fully resolve this longstanding open problem.