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, Mo(G), is defined as Mo(G)=∑uv∈E(G)|nu(e|G)−nv(e|G)|, where nu(e|G) and nv(e|G) represent the number of vertices closer to vertex u than v and closer to v than u, respectively, for an edge e=uv. 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 c≥3. 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.