{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,12,8]],"date-time":"2025-12-08T22:45:44Z","timestamp":1765233944279,"version":"build-2065373602"},"reference-count":26,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2024,12,30]],"date-time":"2024-12-30T00:00:00Z","timestamp":1735516800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"name":"National Science Foundation of China","award":["11561056","2022-ZJ-924"],"award-info":[{"award-number":["11561056","2022-ZJ-924"]}]},{"name":"National Science Foundation of Qinghai Province","award":["11561056","2022-ZJ-924"],"award-info":[{"award-number":["11561056","2022-ZJ-924"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>Let G be a finite simple graph and let A(G) be its adjacency matrix. Then, G is singular if A(G) is singular. The singularity of graphs is of certain interest in graph theory and algebraic combinatorics. For positive integers ai\u22652, i=1,2,\u2026,6. Insert a1\u22122, a2\u22122, a3\u22122, a4\u22122, a5\u22122 and a6\u22122 vertices in the six edges of the complete graph K4, respectively, then the resulting graph is called the K4 homeomorphic graph, denoted by K(a1,a2,a3,a4,a5,a6). In this paper, we give the necessary and sufficient condition for the singularity of K(a1,a2,a3,a4,a5,a6), and we also show that the probability of a K4 homeomorphic graph K(a1,a2,a3,a4,a5,a6) being a singular graph is equal to 193512.<\/jats:p>","DOI":"10.3390\/axioms14010017","type":"journal-article","created":{"date-parts":[[2024,12,31]],"date-time":"2024-12-31T07:34:19Z","timestamp":1735630459000},"page":"17","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["The Singularity of the K4 Homeomorphic Graph"],"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"}]}],"member":"1968","published-online":{"date-parts":[[2024,12,30]]},"reference":[{"unstructured":"Cvetkovi\u0107, D., Doob, M., and Sachs, H. (1980). Spectra of Graphs-Theory and Application, Academic Press.","key":"ref_1"},{"key":"ref_2","first-page":"141","article-title":"The algebraic multiplicity of the number zero in the spectrum of a bipartite graph","volume":"9","author":"Gutman","year":"1972","journal-title":"Mat. Ves."},{"key":"ref_3","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_4","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"},{"doi-asserted-by":"crossref","unstructured":"Graovac, A., Gutman, I., and Trinajsti\u0107, N. (1977). Topological Approach to the Chemistry of Conjugated Molecules, Springer.","key":"ref_5","DOI":"10.1007\/978-3-642-93069-0"},{"unstructured":"Kulli, V.R. (1989). Singular graphs. Recent Studies in Graph Theory, Vishwa Int. Publications.","key":"ref_6"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"60","DOI":"10.13001\/1081-3810.1182","article-title":"On the nullity of graphs","volume":"16","author":"Cheng","year":"2007","journal-title":"Electron. J. Linear Algebra"},{"key":"ref_8","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_9","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_10","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_11","doi-asserted-by":"crossref","first-page":"1387","DOI":"10.1016\/j.laa.2007.12.007","article-title":"On the nullity of bicyclic graphs","volume":"429","author":"Hu","year":"2008","journal-title":"Linear Algebra Appl."},{"key":"ref_12","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_13","doi-asserted-by":"crossref","first-page":"111","DOI":"10.1007\/s00373-008-0825-5","article-title":"On the nullity of bipartite graphs","volume":"25","author":"Omidi","year":"2009","journal-title":"Graphs Combin."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"212","DOI":"10.1016\/j.laa.2005.06.012","article-title":"On the nullity of unicyclic graphs","volume":"408","author":"Tang","year":"2005","journal-title":"Linear Algebra Appl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.laa.2007.06.017","article-title":"On the null spaces of acyclic and unicyclic singular graphs","volume":"427","author":"Nath","year":"2007","journal-title":"Linear Algebra Appl."},{"key":"ref_16","first-page":"73","article-title":"On singular line graphs of trees","volume":"135","author":"Sciriha","year":"1998","journal-title":"Congr. Numer."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"193","DOI":"10.1016\/S0012-365X(97)00036-8","article-title":"On the construction of graphs of nullity one","volume":"181","author":"Sciriha","year":"1998","journal-title":"Discret. Math."},{"key":"ref_18","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_19","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":"Discret. Math."},{"key":"ref_20","first-page":"23","article-title":"Graph singularity. Graph Theory Notes","volume":"25","author":"Brown","year":"1993","journal-title":"Appl. Math."},{"doi-asserted-by":"crossref","unstructured":"Ma, H., Gao, S., and Zhang, B. (2022). The singularity of four kinds of tricyclic graphs. Symmetry, 14.","key":"ref_21","DOI":"10.3390\/sym14122507"},{"doi-asserted-by":"crossref","unstructured":"Ma, H., Gao, Y., and You, X. (2024). The Singularity of Three Kinds of new Tricyclic Graphs. Symmetry, 16.","key":"ref_22","DOI":"10.3390\/sym16111416"},{"key":"ref_23","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_24","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_25","first-page":"112","article-title":"Singular graphs with dihedral group action","volume":"344","author":"Sltan","year":"2021","journal-title":"Discret. Math."},{"key":"ref_26","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\/1\/17\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,10]],"date-time":"2025-10-10T16:56:51Z","timestamp":1760115411000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/14\/1\/17"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,12,30]]},"references-count":26,"journal-issue":{"issue":"1","published-online":{"date-parts":[[2025,1]]}},"alternative-id":["axioms14010017"],"URL":"https:\/\/doi.org\/10.3390\/axioms14010017","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2024,12,30]]}}}