{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,13]],"date-time":"2026-03-13T00:26:52Z","timestamp":1773361612429,"version":"3.50.1"},"reference-count":19,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T00:00:00Z","timestamp":1739491200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>Inverse topological index problems involve determining whether a graph exists with a given integer as its topological index. One such index, the Mostar index,\u00a0Mo(G), is defined as Mo(G)=\u2211uv\u2208E(G)|nu(e|G)\u2212nv(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\u20131093] proposed the following open problem: \u201cWhich nonnegative integers can be realized as Mostar indices of c-cyclic graphs, for a given positive integer c?\u201d. Subsequently, one of the present authors [On the inverse Mostar index problem for molecular graphs, Trans. Comb. 14 (1) (2024) 65\u201377] 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\u22653. 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.<\/jats:p>","DOI":"10.3390\/sym17020291","type":"journal-article","created":{"date-parts":[[2025,2,14]],"date-time":"2025-02-14T04:48:51Z","timestamp":1739508531000},"page":"291","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Resolving the Open Problem by Proving a Conjecture on the Inverse Mostar Index for c-Cyclic Graphs"],"prefix":"10.3390","volume":"17","author":[{"given":"Liju","family":"Alex","sequence":"first","affiliation":[{"name":"Department of Mathematics, Bishop Chulaparambil Memorial (B.C.M) College, Kottayam 686001, Kerala, India"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2576-160X","authenticated-orcid":false,"given":"Kinkar Chandra","family":"Das","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Sungkyunkwan University, Suwon 16419, Republic of Korea"}]}],"member":"1968","published-online":{"date-parts":[[2025,2,14]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"2995","DOI":"10.1007\/s10910-018-0928-z","article-title":"Mostar index","volume":"56","author":"Martinjak","year":"2018","journal-title":"J. Math. Chem."},{"key":"ref_2","unstructured":"Alex, L., and Indulal, G. (2024). Sharp bounds on additively weighted Mostar index of cacti. Commun. Comb. Optim."},{"key":"ref_3","first-page":"126245","article-title":"Mostar index: Results and perspectives","volume":"404","author":"Ali","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_4","first-page":"161","article-title":"Chemical trees with extremal Mostar index","volume":"85","author":"Deng","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_5","doi-asserted-by":"crossref","unstructured":"Sigarreta, J.M. (2020). Mathematical properties of variable topological indices. Symmetry, 13.","DOI":"10.3390\/sym13010043"},{"key":"ref_6","first-page":"319","article-title":"Extremal bicyclic graphs with respect to Mostar index","volume":"355","author":"Tepeh","year":"2019","journal-title":"Appl. Math. Comput."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"17","DOI":"10.1021\/ja01193a005","article-title":"Structural determination of paraffin boiling points","volume":"69","author":"Wiener","year":"1947","journal-title":"J. Am. Chem. Soc."},{"key":"ref_8","doi-asserted-by":"crossref","unstructured":"Todeschini, R., and Consonni, V. (2009). Molecular Descriptors for Chemoinformatics, Wiley-VCH.","DOI":"10.1002\/9783527628766"},{"key":"ref_9","unstructured":"Trinajsti\u0107, N. (1992). Chemical Graph Theory, CRC Press. [2nd ed.]."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"661","DOI":"10.1214\/aoms\/1177704962","article-title":"On fiducial inference","volume":"32","author":"Fraser","year":"1961","journal-title":"Ann. Math. Stat."},{"key":"ref_11","first-page":"327","article-title":"The sum of all distances in bipartite graphs","volume":"45","author":"Gutman","year":"1995","journal-title":"Math. Slovaca."},{"key":"ref_12","first-page":"65","article-title":"On the inverse Mostar index problem for molecular graphs","volume":"14","author":"Alex","year":"2024","journal-title":"Trans. Comb."},{"key":"ref_13","unstructured":"Alex, L., Indulal, G., and Mulloor, J.J. (2024). On the inverse problem of some bond additive indices. Commun. Comb. Optim."},{"key":"ref_14","first-page":"56","article-title":"Inverse problem for Mostar index of chemical trees and unicyclic graphs","volume":"13","author":"Alex","year":"2024","journal-title":"Palest. J. Math."},{"key":"ref_15","first-page":"127709","article-title":"On the \u03c3t-irregularity and the inverse irregularity problem","volume":"441","author":"Dimitrov","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"ref_16","first-page":"491","article-title":"Inverse problem for sigma index","volume":"79","author":"Gutman","year":"2018","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"9411696","DOI":"10.1155\/2021\/9411696","article-title":"On the inverse problem for some topological indices","volume":"2021","author":"Maji","year":"2021","journal-title":"J. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"609","DOI":"10.1007\/s10910-018-0970-x","article-title":"Inverse problem for Zagreb indices","volume":"57","author":"Yurtas","year":"2019","journal-title":"J. Math. Chem."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"1079","DOI":"10.1007\/s10910-024-01581-0","article-title":"Solving the Mostar index inverse problem","volume":"62","author":"Alizadeh","year":"2024","journal-title":"J. Math. 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