{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T02:34:58Z","timestamp":1760150098688,"version":"build-2065373602"},"reference-count":38,"publisher":"MDPI AG","issue":"10","license":[{"start":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T00:00:00Z","timestamp":1697068800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004242","name":"Princess Nourah bint Abdulrahman University","doi-asserted-by":"publisher","award":["PNURSP2023R231","50110000187","UIDB\/04106\/2020","UIDP\/04106\/2020"],"award-info":[{"award-number":["PNURSP2023R231","50110000187","UIDB\/04106\/2020","UIDP\/04106\/2020"]}],"id":[{"id":"10.13039\/501100004242","id-type":"DOI","asserted-by":"publisher"}]},{"name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","award":["PNURSP2023R231","50110000187","UIDB\/04106\/2020","UIDP\/04106\/2020"],"award-info":[{"award-number":["PNURSP2023R231","50110000187","UIDB\/04106\/2020","UIDP\/04106\/2020"]}]},{"DOI":"10.13039\/501100001871","name":"CIDMA","doi-asserted-by":"publisher","award":["PNURSP2023R231","50110000187","UIDB\/04106\/2020","UIDP\/04106\/2020"],"award-info":[{"award-number":["PNURSP2023R231","50110000187","UIDB\/04106\/2020","UIDP\/04106\/2020"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The smallest set of vertices needed to differentiate or categorize every other vertex in a graph is referred to as the graph\u2019s metric dimension. Finding the class of graphs for a particular given metric dimension is an NP-hard problem. This concept has applications in many different domains, including graph theory, network architecture, and facility location problems. A graph G with order n is known as a Toeplitz graph over the subset S of consecutive collections of integers from one to n, and two vertices will be adjacent to each other if their absolute difference is a member of S. A graph G(Zn) is called a zero-divisor graph over the zero divisors of a commutative ring Zn, in which two vertices will be adjacent to each other if their product will leave the remainder zero under modulo n. Since the local fractional metric dimension problem is NP-hard, it is computationally difficult to identify an optimal solution or to precisely determine the minimal size of a local resolving set; in the worst case, the process takes exponential time. Different upper bound sequences of local fractional metric dimension are suggested in this article, along with a comparison analysis for certain families of Toeplitz and zero-divisor graphs. Furthermore, we note that the analyzed local fractional metric dimension upper bounds fall into three metric families: constant, limited, and unbounded.<\/jats:p>","DOI":"10.3390\/sym15101911","type":"journal-article","created":{"date-parts":[[2023,10,12]],"date-time":"2023-10-12T12:46:13Z","timestamp":1697114773000},"page":"1911","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On Sharp Bounds of Local Fractional Metric Dimension for Certain Symmetrical Algebraic Structure Graphs"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-7856-2861","authenticated-orcid":false,"given":"Amal S.","family":"Alali","sequence":"first","affiliation":[{"name":"Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5998-0053","authenticated-orcid":false,"given":"Shahbaz","family":"Ali","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan"}]},{"given":"Muhammad","family":"Adnan","sequence":"additional","affiliation":[{"name":"Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Kahn Campus, Rahim Yar Khan 64200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8641-2505","authenticated-orcid":false,"given":"Delfim F. M.","family":"Torres","sequence":"additional","affiliation":[{"name":"Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal"}]}],"member":"1968","published-online":{"date-parts":[[2023,10,12]]},"reference":[{"key":"ref_1","first-page":"128037","article-title":"Redefining fractal cubic networks and determining their metric dimension and fault-tolerant metric dimension","volume":"452","author":"Arulperumjothi","year":"2023","journal-title":"Appl. Math. Comput."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"210","DOI":"10.1007\/s40314-023-02351-5","article-title":"Graphs whose mixed metric dimension is equal to their order","volume":"42","author":"Ghalav","year":"2023","journal-title":"Comput. Appl. 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