{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:49:23Z","timestamp":1765212563533,"version":"3.46.0"},"reference-count":35,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T00:00:00Z","timestamp":1765152000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001809","name":"National Science Foundation of China","doi-asserted-by":"publisher","award":["11561056"],"award-info":[{"award-number":["11561056"]}],"id":[{"id":"10.13039\/501100001809","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Science Foundation of Qinghai Province","award":["2022-ZJ-924"],"award-info":[{"award-number":["2022-ZJ-924"]}]},{"name":"PhD Research Startup Fundation of Guangdong University of Science and Technology","award":["GKY-2022BSQD-36"],"award-info":[{"award-number":["GKY-2022BSQD-36"]}]},{"name":"Innovation Strong School Project of Guangdong University of Science and Technology","award":["GKY-2022CQTD-5"],"award-info":[{"award-number":["GKY-2022CQTD-5"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>A graph G is said to be singular if its adjacency matrix is a singular matrix. In this paper, by analyzing the structure of Sachs subgraphs on three types of tricyclic graphs \u03d6(a1,a2,b1,b2,c1,c2), \u03c3(a1,a2,b1,b2,c), and \u03c4(a1,a2,a3,a4) and calculating the determinants of their adjacency matrices, we investigate the necessary and sufficient conditions for the singularity of these three types of tricyclic graphs. Furthermore, we derive that the probabilities of singular graphs among tricyclic graphs \u03d6(a1,a2,b1,b2,c1,c2), \u03c3(a1,a2,b1,b2,c), and \u03c4(a1,a2,a3,a4) are 217512,113256, and 67128, respectively.<\/jats:p>","DOI":"10.3390\/axioms14120904","type":"journal-article","created":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:41:41Z","timestamp":1765212101000},"page":"904","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Singularity Conditions and Probabilities of Tricyclic Graphs"],"prefix":"10.3390","volume":"14","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-5503-4668","authenticated-orcid":false,"given":"Haicheng","family":"Ma","sequence":"first","affiliation":[{"name":"College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China"},{"name":"School of Mathematics and Statistics, Qinghai Minzu University, Xining 810007, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0002-0131-2834","authenticated-orcid":false,"given":"Xianghong","family":"He","sequence":"additional","affiliation":[{"name":"College of General Education, Guangdong University of Science and Technology, Dongguan 523083, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2025,12,8]]},"reference":[{"key":"ref_1","unstructured":"Cvetkovi\u0107, D., Doob, M., and Sachs, H. (1980). Spectra of Graphs-Theory and Application, Academic Press."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"289","DOI":"10.1016\/0022-2860(75)80099-8","article-title":"Graphical studies on the relations between the structure and reactivity of conjugated system: The role of non-bonding molecular orbitals","volume":"28","author":"Gutman","year":"1975","journal-title":"J. Mol. Struct."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"63","DOI":"10.1007\/BF02941924","article-title":"Spektren endlicher Grafen","volume":"21","author":"Collatz","year":"1957","journal-title":"Abh. Math. Sem. Univ. Hamburg"},{"key":"ref_4","first-page":"501","article-title":"On the trees with maximum nullity","volume":"56","author":"Li","year":"2006","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"1293","DOI":"10.1016\/j.laa.2009.04.026","article-title":"On the nullity and the matching number of unicyclic graphs","volume":"431","author":"Guo","year":"2009","journal-title":"Linear Algebra Appl."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"2943","DOI":"10.1016\/j.laa.2009.01.007","article-title":"On the nullity of bipartite graphs","volume":"430","author":"Fan","year":"2009","journal-title":"Linear Algebra Appl."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"1799","DOI":"10.1016\/j.laa.2011.01.006","article-title":"On the nullity of tricyclic graphs","volume":"434","author":"Cheng","year":"2011","journal-title":"Linear Algebra Its Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"1793","DOI":"10.1016\/j.laa.2010.09.040","article-title":"A characterization of Graphs with Rank 4","volume":"434","author":"Chang","year":"2011","journal-title":"Linear Algebra Its Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"4241","DOI":"10.1016\/j.laa.2012.01.021","article-title":"A characterization of Graphs with Rank 5","volume":"436","author":"Chang","year":"2012","journal-title":"Linear Algebra Its Appl."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"35","DOI":"10.1016\/S0012-365X(00)00187-4","article-title":"On the nullity of line graphs of trees","volume":"232","author":"Gutman","year":"2001","journal-title":"Discret. Math."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"276","DOI":"10.1016\/j.dam.2013.09.012","article-title":"Bounds for the Matching Number, the Edge Chromatic Number and the Independence Number of a Graph in terms of Rank","volume":"166","author":"Wang","year":"2014","journal-title":"Discret. Math."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"337","DOI":"10.2478\/ausm-2023-0019","article-title":"A survey of the maximal and the minimal nullity in terms of omega invariant on graphs","volume":"15","author":"Oz","year":"2023","journal-title":"Acta Univ. Sapientiae Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"93","DOI":"10.1016\/j.dam.2022.01.020","article-title":"Relation between the nullity of a graph and its matching number","volume":"313","author":"Zhou","year":"2022","journal-title":"Discrete Appl. Math."},{"key":"ref_14","unstructured":"Wu, Q., and Lu, Y. (2025). Improved bounds on the H-rank of a mixed graph in terms of the matching number and fractional matching number. arXiv."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"171","DOI":"10.1016\/j.dam.2016.07.010","article-title":"Nullity of a graph in terms of the dimension of cycle space and the number of pendent vertices","volume":"215","author":"Ma","year":"2016","journal-title":"Discrete Appl. Math."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"44","DOI":"10.1016\/j.dam.2019.08.018","article-title":"The leaf-free graphs with nullity 2c(G) \u2212 1","volume":"277","author":"Chang","year":"2020","journal-title":"Discrete Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"112786","DOI":"10.1016\/j.disc.2021.112786","article-title":"Graphs with nullity 2c(G) + p(G) \u2212 1","volume":"345","author":"Wang","year":"2022","journal-title":"Discret. Math."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"56","DOI":"10.1016\/j.laa.2021.07.005","article-title":"On connected graphs of order n with girth g and nullity n \u2212 g","volume":"630","author":"Zhou","year":"2021","journal-title":"Linear Algebra Its Appl."},{"key":"ref_19","unstructured":"Xu, S. (2024). A characterization of graphs G with nullity n(G) \u2212 d(G) \u2212 1. arXiv."},{"key":"ref_20","doi-asserted-by":"crossref","first-page":"1347","DOI":"10.1016\/j.laa.2012.09.017","article-title":"A characterization of long graphs of arbitrary rank","volume":"438","author":"Wong","year":"2013","journal-title":"Linear Algebra Appl."},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"451","DOI":"10.13001\/1081-3810.1215","article-title":"A characterization of singular graphs","volume":"16","author":"Sciriha","year":"2007","journal-title":"Electron. J. Linear Algebra"},{"key":"ref_22","doi-asserted-by":"crossref","unstructured":"Banerjee, S. (2025). On Structural Properties and Adjacency Spectrum of Coprime Graph of Integers. arXiv.","DOI":"10.1142\/S179355712550069X"},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"138","DOI":"10.1016\/j.laa.2020.05.007","article-title":"Nonsingular (vertex-weighted) block graphs","volume":"602","author":"Singh","year":"2020","journal-title":"Linear Algebra Appl."},{"key":"ref_24","doi-asserted-by":"crossref","first-page":"112851","DOI":"10.1016\/j.disc.2022.112851","article-title":"Nullity and singularity of a graph in which every block is a cycle","volume":"345","author":"Wong","year":"2022","journal-title":"Discret. Math."},{"key":"ref_25","first-page":"73","article-title":"On singular line graphs of trees","volume":"135","author":"Sciriha","year":"1998","journal-title":"Congr. Numer."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"267","DOI":"10.1016\/j.disc.2006.11.040","article-title":"On nut and core singular fullerenes","volume":"308","author":"Sciriha","year":"2008","journal-title":"Discrete Math."},{"key":"ref_27","unstructured":"AL-Tarimshawy, A. (2018). Singular Graphs. [Ph.D. Thesis, University of East Anglia]."},{"key":"ref_28","doi-asserted-by":"crossref","unstructured":"Ma, H., Gao, S., and Zhang, B. (2022). The singularity of four kinds of tricyclic graphs. Symmetry, 14.","DOI":"10.3390\/sym14122507"},{"key":"ref_29","doi-asserted-by":"crossref","unstructured":"Ma, H., Gao, Y., and You, X. (2024). The Singularity of Three Kinds of new Tricyclic Graphs. Symmetry, 16.","DOI":"10.3390\/sym16111416"},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"8949","DOI":"10.3934\/math.2023448","article-title":"The singularity of two kinds of tricyclic graphs","volume":"8","author":"Ma","year":"2023","journal-title":"AIMS Math."},{"key":"ref_31","doi-asserted-by":"crossref","unstructured":"Ma, H. (2025). The Singularity of the K4 Homeomorphic Graph. Axioms, 14.","DOI":"10.3390\/axioms14010017"},{"key":"ref_32","doi-asserted-by":"crossref","first-page":"277","DOI":"10.1080\/10586458.2005.10128928","article-title":"Some computations regarding Foulkes\u2019 conjecture","volume":"14","year":"2005","journal-title":"Exp. Math."},{"key":"ref_33","doi-asserted-by":"crossref","unstructured":"Sharma, M., Nath, R.K., and Shang, Y. (2021). On g-noncommuting graph of a finite group relative to its subgroups. Mathematics, 9.","DOI":"10.3390\/math9233147"},{"key":"ref_34","first-page":"112","article-title":"Singular graphs with dihedral group action","volume":"344","author":"Sltan","year":"2021","journal-title":"Discret. Math."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"379","DOI":"10.1007\/s10801-018-0860-0","article-title":"Remarks on singular Cayley graphs and vanishing elements of simple groups","volume":"50","author":"Siemons","year":"2019","journal-title":"J. Algebr. Comb."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/904\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T16:45:01Z","timestamp":1765212301000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/12\/904"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,12,8]]},"references-count":35,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2025,12]]}},"alternative-id":["axioms14120904"],"URL":"https:\/\/doi.org\/10.3390\/axioms14120904","relation":{},"ISSN":["2075-1680"],"issn-type":[{"value":"2075-1680","type":"electronic"}],"subject":[],"published":{"date-parts":[[2025,12,8]]}}}