{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,7,3]],"date-time":"2026-07-03T16:13:52Z","timestamp":1783095232093,"version":"3.54.6"},"reference-count":20,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T00:00:00Z","timestamp":1754611200000},"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":["RGP.2\/372\/45"],"award-info":[{"award-number":["RGP.2\/372\/45"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A quantitative study of the complicated three-dimensional structures of artificial atoms in the field of intense matter physics requires a collaborative method that combines a statistical analysis of unusual graph features related to atom topology. Simplified circuits can also be produced by using similar transformations to streamline complex circuits that need laborious mathematical calculations during analysis. These modifications can also be used to determine the number of spanning trees required for specific graph families. The explicit derivation of formulas to determine the number of spanning trees for novel pyramid graph types based on the Fritsch graph, which is one of only six graphs in which every neighborhood is a 4- or 5-vertex cycle, is the focus of our study. We conduct this by utilizing our understanding of difference equations, weighted generating function rules, and the strength of analogous transformations found in electrical circuits.<\/jats:p>","DOI":"10.3390\/axioms14080622","type":"journal-article","created":{"date-parts":[[2025,8,8]],"date-time":"2025-08-08T15:30:52Z","timestamp":1754667052000},"page":"622","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["The Complexity of Classes of Pyramid Graphs Based on the Fritsch Graph and Its Related Graphs"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0009-0008-5943-1290","authenticated-orcid":false,"given":"Ahmad","family":"Asiri","sequence":"first","affiliation":[{"name":"Department of Mathematics, Applied College at Mahail Aseer, King Khalid University, Abha 61421, Saudi Arabia"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3809-2521","authenticated-orcid":false,"given":"Salama Nagy","family":"Daoud","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Nunawara 30001, Saudi Arabia"},{"name":"Department of Mathematics and Computer Sciences, Faculty of Science, Menoufia University, Shebin El Kom 32511, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2025,8,8]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"41","DOI":"10.1016\/j.tcs.2014.01.012","article-title":"Counting spanning trees using modular decomposition","volume":"526","author":"Nikolopoulos","year":"2014","journal-title":"Theor. Comput. Sci."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"9424605","DOI":"10.1155\/2019\/9424605","article-title":"Expanding network analysis tools in psychological networks: Minimal spanning trees, participation coefficients, and motif analysis applied to a network of 26 psychological attributes","volume":"2019","author":"Letina","year":"2019","journal-title":"Complexity"},{"key":"ref_3","first-page":"264","article-title":"Asymptotic enumeration theorems for the number of spanning trees and Eulerian trail in circulant digraphs & graphs","volume":"43","author":"Zhang","year":"1999","journal-title":"Sci. China Ser. A Math."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"59","DOI":"10.1016\/0166-218X(95)00118-B","article-title":"On the number of spanning trees and Eulerian torus in iterated line digraph","volume":"73","author":"Huaxiao","year":"1997","journal-title":"Discret. App. Math."},{"key":"ref_5","unstructured":"Applegate, D.L., Bixby, R.E.V., and Cook, W.J. (2006). The Traveling Salesman Problem: A Computational Study, Princeton University Press."},{"key":"ref_6","first-page":"601","article-title":"Network reliability analysis by counting the number of spanning trees, ISCIT 2004","volume":"1","author":"Atajan","year":"2004","journal-title":"IEEE Int. Symp. Commun. Inf. Technol."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"5967604","DOI":"10.1155\/2018\/5967604","article-title":"Approximate Method to Evaluate Reliability of Complex Networks","volume":"2018","year":"2018","journal-title":"Complexity"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1002\/andp.18471481202","article-title":"\u00dcber die Aufl\u00f6sung der Gleichungen auf welche man bei der Untersucher der linearen Verteilung galuanischer Strome gefhrt wird","volume":"72","author":"Kirchhoff","year":"1847","journal-title":"Ann. Phg. Chem."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"197","DOI":"10.1016\/0095-8956(74)90065-3","article-title":"A certain polynomial of a graph and graphs with an extremal number of trees","volume":"16","author":"Kelmans","year":"1974","journal-title":"J. Comb. Theory B"},{"key":"ref_10","unstructured":"Biggs, N.L. (1993). Algebraic Graph Theory, Cambridge University Press. [2nd ed.]."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"217","DOI":"10.1140\/epjp\/i2015-15217-y","article-title":"The Deletion-Contraction Method for Counting the Number of Spanning Trees of Graphs","volume":"130","author":"Daoud","year":"2015","journal-title":"Eur. J. Phys. Plus"},{"key":"ref_12","first-page":"85","article-title":"Number of Spanning Trees in Different Product of Complete and Complete Tripartite Graphs","volume":"139","author":"Daoud","year":"2018","journal-title":"Ars Comb."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"71142","DOI":"10.1109\/ACCESS.2019.2917535","article-title":"Number of Spanning Trees of Cartesian and Composition Products of Graphs and Chebyshev Polynomials","volume":"7","author":"Daoud","year":"2019","journal-title":"IEEE Access"},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"El Deen, M.R.Z., Aboamer, W.A., and El-Sherbiny, H.M. (2023). The Complexity of the Super Subdivision of Cycle-Related Graphs Using Block Matrices. Computation, 11.","DOI":"10.3390\/computation11080162"},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"441","DOI":"10.1016\/j.laa.2009.08.028","article-title":"Determinant identities for Laplace matrices","volume":"432","author":"Teufl","year":"2010","journal-title":"Linear Algebra Appl."},{"key":"ref_16","first-page":"39","article-title":"Number of spanning Trees in the sequence of some Nonahedral graphs","volume":"115","author":"Daoud","year":"2020","journal-title":"Util. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1088\/0305-4470\/10\/6\/004","article-title":"Number of spanning trees on a lattice","volume":"10","author":"Wu","year":"1977","journal-title":"J. Phys. A Math. Gen."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"491","DOI":"10.1017\/S096354830500684X","article-title":"Asymptotic enumeration of spanning trees","volume":"14","author":"Lyons","year":"2005","journal-title":"Combin. Probab. Comput."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"206","DOI":"10.1016\/j.dam.2014.01.015","article-title":"The number of spanning trees in Apollonian networks","volume":"169","author":"Zhang","year":"2014","journal-title":"Discret. Appl. Math."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"415001","DOI":"10.1088\/1751-8113\/43\/41\/415001","article-title":"On the number of spanning trees on various lattices","volume":"43","author":"Teufl","year":"2010","journal-title":"J. Phys. A Math. 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