{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,7]],"date-time":"2026-02-07T03:01:32Z","timestamp":1770433292028,"version":"3.49.0"},"reference-count":28,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T00:00:00Z","timestamp":1770336000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100007446","name":"King Khalid University","doi-asserted-by":"publisher","award":["RGP.2\/229\/46"],"award-info":[{"award-number":["RGP.2\/229\/46"]}],"id":[{"id":"10.13039\/501100007446","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Complexity (number of spanning trees) is an essential and significant component in the design of communication networks (graphs). To ensure strong resistance and stiffness and to enhance the probability of a connection between two vertices, improvements to a network\u2019s quality and perfection increase the number of trees that span it. Using block matrices and linear algebra techniques, we derive explicit formulas for the number of spanning trees of new graph families that are produced from star graphs in this study. The number of spanning trees in a graph is measured by the entropy of spanning trees, also known as asymptotic complexity, a graph theory metric that assesses the network\u2019s structural robustness and dependability. Increased flexibility, stronger diverse connections, and improved resistance to random structural changes are all indicated by higher entropy. We also investigate the entropy of spanning trees on our graphs at the end of this study. Lastly, we compare the entropy of our graphs to that of other previously studied graphs with average degrees of four and five.<\/jats:p>","DOI":"10.3390\/axioms15020122","type":"journal-article","created":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T13:33:50Z","timestamp":1770384830000},"page":"122","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["On the Number of Spanning Trees of New Graph Families Created from the Star Graph and the Examination of Their Entropies"],"prefix":"10.3390","volume":"15","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-3809-2521","authenticated-orcid":false,"given":"Salama Nagy","family":"Daoud","sequence":"first","affiliation":[{"name":"Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawara 41411, Saudi Arabia"},{"name":"Department of Mathematics and Computer Sciences, Faculty of Science, Menoufia University, Shebin El Kom 32511, Egypt"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-5943-1290","authenticated-orcid":false,"given":"Ahmad","family":"Asiri","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Applied College at Mahail Aseer, King Khalid University, Abha 61421, Saudi Arabia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2026,2,6]]},"reference":[{"key":"ref_1","unstructured":"Cvetkovi\u0115, D., Doob, M., and Sachs, H. (1995). Spectra of Graphs: Theory and Applications, Johann Ambrosius Barth. [3rd ed.]."},{"key":"ref_2","unstructured":"Applegate, D.L., Bixby, R.E., Chv\u00e1tal, V., and Cook, W.J. (2006). The Traveling Salesman Problem: A Computational Study, Princeton University Press."},{"key":"ref_3","first-page":"957","article-title":"Application of Graph Theory in Computer Science and Engineering","volume":"104","author":"Singh","year":"2014","journal-title":"Int. J. Comput. Appl."},{"key":"ref_4","unstructured":"Deo, N. (1974). Graph Theory with Applications to Engineering and Computer Science, Prentice-Hall."},{"key":"ref_5","unstructured":"Dolan, A.K., and Aldous, J. (1993). Networks and Algorithms: An Introductory Approach, Wiley-Interscience."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"497","DOI":"10.1002\/andp.18471481202","article-title":"Uber die Auflosung der Gleichungen, auf welche man be ider Untersuchung der Linearen Verteilung galvanischer Storme gefuhrt wird","volume":"72","author":"Kirchhoff","year":"1847","journal-title":"Ann. Phys. Chem."},{"key":"ref_7","first-page":"276","article-title":"A Theorm on Trees","volume":"23","author":"Cayley","year":"1889","journal-title":"Quart. J. Math."},{"key":"ref_8","first-page":"50","article-title":"On the enumeration of multipartite spanning trees of the complete graph","volume":"38","author":"Clark","year":"2003","journal-title":"Bull. ICA"},{"key":"ref_9","unstructured":"Sedlacek, J. (1970). Lucas number in graph theory. Mathematics (Geometry and Graph Theory) (Chech), Univerzita Karlova."},{"key":"ref_10","unstructured":"Guy, M., Saver Hanani, N., and Schonheim, J. (1970). On the Skeleton of a Graph or Digraph. Combinatorial Structures and Their Applications, Gordon and Breach."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"229","DOI":"10.1080\/00207168708803568","article-title":"The number of spanning trees in a Prism","volume":"21","author":"Boesch","year":"1987","journal-title":"Inter. J. Comput. Math."},{"key":"ref_12","first-page":"820549","article-title":"On a class of some pyramid graphs and Chebyshev polynomials","volume":"2013","author":"Daoud","year":"2013","journal-title":"J. Math. Probl. Eng. Hindawi Publ. Corp."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"205","DOI":"10.1016\/j.jtusci.2016.04.002","article-title":"The complexity of Some Families of Cycle-Related Graphs","volume":"11","author":"Daoud","year":"2017","journal-title":"J. Taibah Univ. Sci."},{"key":"ref_14","doi-asserted-by":"crossref","unstructured":"MEl Deen, 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":"197","DOI":"10.1016\/0095-8956(74)90065-3","article-title":"A certain polynomial of a graph and graphs with an extermal number of trees","volume":"16","author":"Kelmans","year":"1974","journal-title":"J. Comb. Theory"},{"key":"ref_16","first-page":"742","article-title":"Neuer Beweis eines Satzes \u00fcber Permutationen","volume":"27","year":"1918","journal-title":"Arch. Math. Phys."},{"key":"ref_17","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_18","doi-asserted-by":"crossref","first-page":"641","DOI":"10.1515\/math-2016-0055","article-title":"On the number of spanning trees, the Laplacian eigenvalues, and the Laplacian Estrada index of subdivided-line graphs","volume":"14","author":"Shang","year":"2016","journal-title":"Open Math"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Asiri, A., and Daoud, S.N. (2025). Enumerating the Number of Spanning Trees of Pyramid Graphs. Axioms, 14.","DOI":"10.3390\/axioms14030148"},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Asiri, A., and Daoud, S.N. (2025). The Complexity of Classes of Pyramid Graphs Based on the Fritsch Graph and Its Related Graphs. Axioms, 14.","DOI":"10.3390\/axioms14080622"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"191","DOI":"10.1007\/BF01788093","article-title":"Spanning tree Formulas and Chebyshev Polynomials","volume":"2","author":"Boeschand","year":"1986","journal-title":"J. Graphs Comb."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"334","DOI":"10.1016\/j.disc.2004.10.025","article-title":"Chebyshev polynomials and spanning trees formulas for circulant and related graphs","volume":"298","author":"Zhang","year":"2005","journal-title":"Discret. Math."},{"key":"ref_23","unstructured":"Marcus, M. (1964). A Servy of Matrix Theory and Matrix Inequalities, Allyn and Bacon Inc."},{"key":"ref_24","unstructured":"Colbourn, C.J. (1980). The Combinatorics of Network Reliability, Oxford University Press."},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"417","DOI":"10.1002\/net.3230210404","article-title":"Uniformly-most reliable networks do not always exist","volume":"21","author":"Myrvold","year":"1991","journal-title":"Networks"},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"016116","DOI":"10.1103\/PhysRevE.83.016116","article-title":"Spanning trees in a fractal scale\u2014Free lattice","volume":"83","author":"Zhang","year":"2011","journal-title":"Phys. Rev. E"},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"649","DOI":"10.1007\/s10955-006-9262-0","article-title":"Spanning trees on the Sierpinski gasket","volume":"126","author":"Chang","year":"2007","journal-title":"J. Stat. Phys."},{"key":"ref_28","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."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/122\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,2,6]],"date-time":"2026-02-06T13:46:50Z","timestamp":1770385610000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/15\/2\/122"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,6]]},"references-count":28,"journal-issue":{"issue":"2","published-online":{"date-parts":[[2026,2]]}},"alternative-id":["axioms15020122"],"URL":"https:\/\/doi.org\/10.3390\/axioms15020122","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,6]]}}}