{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:54:25Z","timestamp":1760147665477,"version":"build-2065373602"},"reference-count":48,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T00:00:00Z","timestamp":1676505600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"Deanship of Scientific Research at King Khalid University","award":["R.G.P.1\/277\/43"],"award-info":[{"award-number":["R.G.P.1\/277\/43"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>The metric dimension has various applications in several fields, such as computer science, image processing, pattern recognition, integer programming problems, drug discovery, and the production of various chemical compounds. The lowest number of vertices in a set with the condition that any vertex can be uniquely identified by the list of distances from other vertices in the set is the metric dimension of a graph. A resolving function of the graph G is a map \u03d1:V(G)\u2192[0,1] such that \u2211u\u2208R{v,w}\u03d1(u)\u22651, for every pair of adjacent distinct vertices v,w\u2208V(G). The local fractional metric dimension of the graph G is defined as ldimf(G) = min{\u2211v\u2208V(G)\u03d1(v), where \u03d1 is a local resolving function of G}. This paper presents a new family of planar networks namely, rotationally heptagonal symmetrical graphs by means of up to four cords in the heptagonal structure, and then find their upper-bound sequences for the local fractional metric dimension. Moreover, the comparison of the upper-bound sequence for the local fractional metric dimension is elaborated both numerically and graphically. Furthermore, the asymptotic behavior of the investigated sequences for the local fractional metric dimension is addressed.<\/jats:p>","DOI":"10.3390\/sym15020530","type":"journal-article","created":{"date-parts":[[2023,2,16]],"date-time":"2023-02-16T03:32:30Z","timestamp":1676518350000},"page":"530","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":7,"title":["On Rotationally Symmetrical Planar Networks and Their Local Fractional Metric Dimension"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5998-0053","authenticated-orcid":false,"given":"Shahbaz","family":"Ali","sequence":"first","affiliation":[{"name":"Department of Mathematics, The Islamia University of Bahawalpur, Rahim Yar Khan Campus, Rahim Yar Khan 64200, Pakistan"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7080-3824","authenticated-orcid":false,"given":"Rashad","family":"Ismail","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science and Arts, King Khalid University, Muhayl Assir 61913, Saudi Arabia"},{"name":"Department of Mathematics and Computer, Faculty of Science, IBB University, IBB 70270, Yemen"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-6037-6769","authenticated-orcid":false,"given":"Francis Joseph","family":"H. Campena","sequence":"additional","affiliation":[{"name":"Department of Mathematics and Statistics, College of Science, De La Salle University, 2401 Taft Avenue, Manila 1004, Philippines"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-5162-2692","authenticated-orcid":false,"given":"Hanen","family":"Karamti","sequence":"additional","affiliation":[{"name":"Department of Computer Sciences, College of Computer and Information Sciences, Princess Nourah Bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi Arabia"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-9916-2031","authenticated-orcid":false,"given":"Muhammad Usman","family":"Ghani","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Khawaja Fareed University of Engineering and Information Technology, Rahim Yar Khan 64200, Pakistan"}]}],"member":"1968","published-online":{"date-parts":[[2023,2,16]]},"reference":[{"key":"ref_1","first-page":"445","article-title":"Dominating and reference sets in a graph","volume":"22","author":"Slater","year":"1988","journal-title":"J. Math. Phys. Sci."},{"key":"ref_2","first-page":"191","article-title":"On the metric dimension of a graph","volume":"2","author":"Melter","year":"1976","journal-title":"Ars Combin"},{"key":"ref_3","first-page":"295","article-title":"On the metric dimension of convex polytopes","volume":"10","author":"Imran","year":"2013","journal-title":"AKCE Int. J. Graphs Comb."},{"key":"ref_4","first-page":"111","article-title":"On the metric dimension of rotationally-symmetric graphs","volume":"124","author":"Imran","year":"2016","journal-title":"Ars Comb."},{"key":"ref_5","first-page":"21","article-title":"Families of regular graphs with constant metric dimension","volume":"75","author":"Javaid","year":"2008","journal-title":"Util. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1016\/0166-218X(95)00106-2","article-title":"Landmarks in graphs","volume":"70","author":"Khuller","year":"1996","journal-title":"Discret. Appl. Math."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1016\/0734-189X(84)90051-3","article-title":"Metric bases in digital geometry","volume":"25","author":"Melter","year":"1984","journal-title":"Comput. Vis. Graph. Image Process."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1287\/moor.1030.0070","article-title":"On metric generators of graphs","volume":"29","author":"Sebo","year":"2004","journal-title":"Math. Oper. Res."},{"key":"ref_9","unstructured":"Beerliova, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihal\u00e1k, M., and Ram, L.S. (2005). International Workshop on Graph-Theoretic Concepts in Computer Science, Springer."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"20","DOI":"10.1016\/j.jda.2006.09.002","article-title":"On minimum metric dimension of honeycomb networks","volume":"6","author":"Manuel","year":"2008","journal-title":"J. Discret. Algorithms"},{"key":"ref_11","first-page":"133","article-title":"Metric dimension of hexogonal cellular networks","volume":"4","author":"Shreedhar","year":"2010","journal-title":"Int. J. Math. Sci. Engg. Appl."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1","DOI":"10.1016\/j.jda.2013.09.002","article-title":"On the metric dimension of HDN","volume":"26","author":"Xu","year":"2014","journal-title":"J. Discret. Algorithms"},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"99","DOI":"10.1016\/S0166-218X(00)00198-0","article-title":"Resolvability in graphs and the metric dimension of a graph","volume":"105","author":"Chartrand","year":"2000","journal-title":"Discret. Appl. Math."},{"key":"ref_14","first-page":"157","article-title":"The metric dimension and metric independence of a graph","volume":"39","author":"Currie","year":"2001","journal-title":"J. Comb. Math. Comb. Comput."},{"key":"ref_15","unstructured":"Benish, H., Murtaza, M., and Javaid, I. (2018). The Fractional Local Metric Dimension of Graphs. arXiv."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"172329","DOI":"10.1109\/ACCESS.2020.3025018","article-title":"Sharp bounds of local fractional metric dimensions of connected networks","volume":"8","author":"Javaid","year":"2020","journal-title":"IEEE Access"},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"82404","DOI":"10.1109\/ACCESS.2020.2991685","article-title":"Local fractional metric dimensions of rotationally symmetric and planar networks","volume":"8","author":"Liu","year":"2020","journal-title":"IEEE Access"},{"key":"ref_18","unstructured":"Chartrand, G., Lesniak, L., and Zhang, P. (1996). Graphs & Digraphs, Chapman & Hall."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Gross, J.L., Yellen, J., and Anderson, M. (2018). Graph Theory and Its Applications, Chapman and Hall\/CRC.","DOI":"10.1201\/9780429425134"},{"key":"ref_20","unstructured":"West, D.B. (2001). Introduction to Graph Theory, Prentice hall."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1023\/A:1025745406160","article-title":"On k-dimensional graphs and their bases","volume":"46","author":"Buczkowski","year":"2003","journal-title":"Period. Math. Hung."},{"key":"ref_22","first-page":"549","article-title":"Leaves of trees","volume":"14","author":"Slater","year":"1975","journal-title":"Congr. Numer."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1002\/net.3230170105","article-title":"Domination and location in acyclic graphs","volume":"17","author":"Slater","year":"1987","journal-title":"Networks"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Aisyah, S., Utoyo, M.I., and Susilowati, L. (2020). The Fractional Local Metric Dimension of Comb Product Graphs. Baghdad Sci. J., 17.","DOI":"10.21123\/bsj.2020.17.4.1288"},{"key":"ref_25","first-page":"371","article-title":"On the metric dimension of the Jahangir graph","volume":"50","author":"Tomescu","year":"2007","journal-title":"Bulletin Math\u00e9matique de la Soci\u00e9t\u00e9 des Sciences Math\u00e9matiques de Roumanie"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"31","DOI":"10.1016\/j.disc.2005.09.015","article-title":"The metric dimension of Cayley digraphs","volume":"306","author":"Fehr","year":"2006","journal-title":"Discret. Math."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"1584","DOI":"10.1016\/j.disc.2011.05.039","article-title":"The fractional metric dimension of graphs","volume":"312","author":"Arumugam","year":"2012","journal-title":"Discret. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"3571","DOI":"10.1016\/j.amc.2010.08.038","article-title":"A note on the partition dimension of Cartesian product graphs","volume":"217","author":"Yero","year":"2010","journal-title":"Appl. Math. Comput."},{"key":"ref_29","first-page":"793","article-title":"A paradigmatic approach to investigate restricted totient graphs and their indices","volume":"16","author":"Ali","year":"2021","journal-title":"Comput. Sci."},{"key":"ref_30","doi-asserted-by":"crossref","unstructured":"Ali, S., Mahmood, M.K., and Mateen, M.H. (2019, January 1\u20132). New labeling algorithm on various classes of graphs with applications. Proceedings of the 2019 International Conference on Innovative Computing (ICIC), Lahore, Pakistan.","DOI":"10.1109\/ICIC48496.2019.8966729"},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ali, S., Falc\u00f3n, R.M., and Mahmood, M.K. (2021). Local fractional metric dimension of rotationally symmetric planar graphs arisen from planar chorded cycles. arXiv.","DOI":"10.1155\/2021\/6613033"},{"key":"ref_32","unstructured":"Yero, I.G. (2013). On the strong partition dimension of graphs. arXiv."},{"key":"ref_33","doi-asserted-by":"crossref","first-page":"67","DOI":"10.1007\/s40840-017-0463-2","article-title":"Zagreb indices and multiplicative zagreb indices of eulerian graphs","volume":"42","author":"Liu","year":"2019","journal-title":"Bull. Malays. Math. Sci. Soc."},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"1950135","DOI":"10.1142\/S0218348X19501354","article-title":"The Hosoya index of graphs formed by a fractal graph","volume":"27","author":"Liu","year":"2019","journal-title":"Fractals"},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"4271783","DOI":"10.1155\/2019\/4271783","article-title":"Number of spanning trees in the sequence of some graphs","volume":"2019","author":"Liu","year":"2019","journal-title":"Complexity"},{"key":"ref_36","first-page":"23","article-title":"A novel labeling algorithm on several classes of graphs","volume":"49","author":"Mahmood","year":"2017","journal-title":"Punjab Univ. J. Math."},{"key":"ref_37","first-page":"29","article-title":"On super totient numbers, with applications and algorithms to graph labeling","volume":"143","author":"Mahmood","year":"2019","journal-title":"Ars Comb."},{"key":"ref_38","first-page":"61","article-title":"New numbers on euler\u2019s totient function with applications","volume":"14","author":"Ali","year":"2019","journal-title":"J. Math. Ext."},{"key":"ref_39","first-page":"1","article-title":"Novel classes of integers and their applications in graph labeling","volume":"1","author":"Ali","year":"2021","journal-title":"Hacet. J. Math. Stat."},{"key":"ref_40","doi-asserted-by":"crossref","first-page":"3761","DOI":"10.3934\/math.2021223","article-title":"A paradigmatic approach to investigate restricted hyper totient graphs","volume":"6","author":"Ali","year":"2021","journal-title":"AIMS Math."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.dam.2014.01.006","article-title":"On the fractional metric dimension of graphs","volume":"170","author":"Feng","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"ref_42","unstructured":"Feng, M., and Wang, K. (2012). On the fractional metric dimension of corona product graphs and lexicographic product graphs. arXiv."},{"key":"ref_43","doi-asserted-by":"crossref","unstructured":"Liu, J.B., Kashif, A., Rashid, T., and Javaid, M. (2019). Fractional metric dimension of generalized Jahangir graph. Mathematics, 7.","DOI":"10.3390\/math7010100"},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"012043","DOI":"10.1088\/1755-1315\/243\/1\/012043","article-title":"On the local fractional metric dimension of corona product graphs","volume":"243","author":"Aisyah","year":"2019","journal-title":"IOP Conf. Ser. Earth Environ. Sci."},{"key":"ref_45","doi-asserted-by":"crossref","first-page":"608","DOI":"10.1016\/j.ins.2020.09.050","article-title":"Determination of journeys order based on graph\u2019s Wiener absolute index with bipolar fuzzy information","volume":"545","author":"Poulik","year":"2021","journal-title":"Inf. Sci."},{"key":"ref_46","doi-asserted-by":"crossref","first-page":"4557","DOI":"10.1007\/s10462-021-10111-2","article-title":"Estimation of most effected cycles and busiest network route based on complexity function of graph in fuzzy environment","volume":"55","author":"Poulik","year":"2022","journal-title":"Artif. Intell. Rev."},{"key":"ref_47","doi-asserted-by":"crossref","first-page":"239","DOI":"10.21136\/MB.2010.140702","article-title":"The local metric dimension of a graph","volume":"135","author":"Okamoto","year":"2010","journal-title":"Math. Bohem."},{"key":"ref_48","doi-asserted-by":"crossref","first-page":"124","DOI":"10.1016\/j.ejc.2014.11.003","article-title":"Distinguishing graphs by edge-colourings","volume":"45","author":"Kalinowski","year":"2015","journal-title":"Eur. J. Comb."}],"container-title":["Symmetry"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/530\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T18:37:46Z","timestamp":1760121466000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2073-8994\/15\/2\/530"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,2,16]]},"references-count":48,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2023,2]]}},"alternative-id":["sym15020530"],"URL":"https:\/\/doi.org\/10.3390\/sym15020530","relation":{},"ISSN":["2073-8994"],"issn-type":[{"type":"electronic","value":"2073-8994"}],"subject":[],"published":{"date-parts":[[2023,2,16]]}}}